PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME VII
CONVEXITY
AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAN...
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PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME VII
CONVEXITY
AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND
1963
Prepared by the American Mathematical Society under Contract Number AF 49 (638) — 964 with the United States Air Force
Copyright 1963 by the American Mathematical Society Printed in the United States of America
All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers
CONTENTS PREFACE
•
ix
fl4TRODUCTION
On systems of linear inequalities in Hermitian matrix variables By RICHARD BELLMAN and
.
.
.
1
Fui
Minimum area of a set of constant width
13
By A. S. BESICOVITCH
On semicircles inscribed into sets of constant width
15
By A. S. BEsIcovITca
A cage to hold a unit.sphere
19
By A. S. BESICOVITCH
On singular points of convex surfaces
21
By A. S. BESICOVITCH
On the set of directions of linear segments on a convex surface By A. S. BESIC0VITCH
The support functionals of a convex set By ERRErr
and R. R. PHELPS
Topological classification of convex sets
37
By HARRY CORSON and VICTOR KLEE
An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size
53
By H. S. M. COXETER
Rotundity By D. F. CumA
73
A characterization of the circle
99
By LUDWIG W. DANZER
Helly's theorem and its relatives
101
By LUDWIG DANZER, BRANKO GRUNBAUM and VICToR KLEE
An extremal problem for plane convex curves
181
By CHANDLER DAViS
Notions generalizing convexity for functions defined on spaces of matrices
187
By CHANDLER DAVIS
Some near.sphericity results
203
By ARYEH DYORETZKY
On the Krein-Milman theorem By On
211
FAN
Lipschitzian mappings of convex bodies
221
By DAVID GALE
Neighborly and cyclic polytopes
225
By DAVID GALE Measures of symmetry
233
for convex sets
By BRANKO GRUNBAUM
V
271
Borsuk's problem and related questions By BRANKO GRUNBAUM
On polyhedral graphs
.
By BRANKO GRUNBAUM and THEODORE S. MOTZKIN
Convex curves of constant Minkowski breadth
291
By PRESTON C. HAMMER
Semispaces and the topology of convexity
305
By PaEs!roN C. HAMMER
On simple linear programming problems
317
By A. J. HOFFMAN
Total positivity and convexity preserving transformations .
.
.
. 329
By SAMUEL KARLIN
Infinite-dimensional intersection theorems
349
By VICTOR KLEE
Endovectors
361
By T. S. MoTZKIN
Representation of points of a set as linear combinations of boundary points
389
By T. S. MOTZKIN and E. G. STIWJS
Support cones and their generalizations
393
By R. R. PHELPS
Convex spaces associated with a family of linear inequalities.
.
. 403
.
. 437
By H. PORITSKY
A combinatorial lemma on the existence of convex means and its application to weak compactness
.
By VLASTIMIL PTAK
Convex cones and spectral theory
451
By HELMUT H. SCHAEFER
The dual cone and Helly type theorems
473
By F. A. VALENTINE UNSOLVED PROBLEMS INDEX OF UNSOLVED PROBLEMS AUTHOR INDEX SUBJECT INDEX
495 501 503
509
PREFACE
Of the thirty-two papers in this volume, seventeen were presented at the Symposium on Convexity and the others were submitted later. (Symposium speakers were Besicovitch, Coxeter, Danzer, Davis, Day, Dvoretzky, Fan, Gale, GrUnbaum, Hammer, Hoffman, Karlin, Klee, Motzkin, Phelps, Pták, Schaefer, and Valentine.) The thirty-third "paper" included here is a report on unsolved problems, based on the Symposium's session devoted to them, on informal discussions during the Symposium, and on later communications from the participants. The papers are arranged alphabetically by author, since this seems most convenient for reference purposes. Interrelationships of the various papers, and their relation to the theory as a whole, are discussed in the Introduction. Since some of the individual bibliographies were so long and in such a state of flux, a common list of references did not seem feasible. However, the Author Index (in conjunction with the individual bibliographies) should be a fair substitute for such a list, and also makes it easy to learn which of the papers cite the work of a given author. There are also a Subject Index and an Index of Unsolved Problems. The editor is indebted to Professors Gale and Grimbaum for their assistance planning the Symposium, to Dr. PtIk and Professors Besicovitch, Coxeter, Day, Fan, and Motzkin for presiding at Symposium sessions, and to Dr. Danzer and Professors Besicovitch, Corson, Firey, Grünbaum, McMinn, and Motzkin for refereeing some of the papers. In particular, the advice and assistance of Branko Grünbaum have been invaluable. The details of publication have been capably handled by Miss Ellen Swanson, Head of the American Mathematical Society's Editorial Department.
Victor Klee
VII
INTRODUCTION
The systematic study of convex sets was initiated by H. Brunn and H. For most of the important notions in the field, at least a germ
Minkowski.
can be found in the latter's collected works (1911). Not only does the theory of convexity play a central role in Minkowslçi's geometry of numbers, but it of elementary number theory. nontechnical also shares some of and of strong intuitive appeal. The are Its basic subject is primarily one \of ideas rather than machinery, and does not lend itself readily to unified treatment. It abounds in attractive special problems, and many mathematicians working mainly in other fields have published one or two papers on convexity. These aspects have accounted for the rapid the theory. but disorganized growth The 1934 by T. Bonnesen and W. Fenchel was an excellent summary of a body of material, and is still a standard source of information in the field. Though in coverage, they cited moçe than 450 references; a current survey of the sAMe degree of completeness be a tremendous undertaking, probably not feasible. More than half of their book emphasized various quantitative notions such as diamet4r, area, volume and mixed volumes.
Since 1934 these same notions have contFnued to play an important role. However, more striking (since less predictable) has been the intensive development of several qualitative aspects of the including the combinatorial geometry associated with intersection and covering properties, the refinement and application (especially in functional analysis and game theory) of such notions as extremal separation properties, the study of convexity in infinite-dimensional spaces, increasing use of convexity as a descriptive tool, and the evolution of various analogues generalizations of convexity. Though several quantitative investigations are included here, the Symposium was intended primarily to emphasize the more qualitative aspects of the theory. In particular, the five aspects listed above are all represented in the present volume. Among the unavoidable omissions, two are especially regretted by the editor. There is nothing here about the geometry of convex surfaces and the associated development of metric methods in differential geometry, carried out by A. D. Aleksandrov and his students in the Soviet Union and in this country by H. Busemann. Also omitted are the important results on infinitedimensional simplexes, boundaries and extremal structure which have been developed in the past few years by G. Choquet and others. In addition to the wide range of topics treated here, there is much variety of approach. Some of the shorter papers treat a single problem in full detail, while at the other extreme are several long papers which include very few proofs but survey broad areas in the field of convexity.
Four of the papers are set in the Euclidean plane BESICOVITCH's first paper gives a short proof of the known fact that a set of given constant width lx
INTRODUCTION
has minimum area when it is a Reuleaux triangle. His second paper solves affirmatively a special case of the following problem: Must a set of constant width w contain a semicircle of diameter w? DANZER gives a short proof of the known result that if C is a dosed convex curve in E2 which does not contain exactly three vertices of any rectangle, then C is a circle. In his first paper, DAVIS characterizes rectangles by means of an extremal area property involving inscribed crosses and also discusses a related conjecture of Ungar on extremal perimeters. HAMMER's first paper is set in an arbitrary Minkowski plane where by
the use of outwardly simple line families he is able to give an analytic representation for all convex curves of constant Minkowski width. He also summarizes his earlier work on diametral lines and associated convex bodies. BESICOVITCH's third paper discusses Coxeter's problem of finding the smallest cage (edges of a convex polyhedron) which will hold a unit-sphere in B3 without permitting it to escape. His other two papers give new proofs of known results concerning smoothness properties of a convex body K in E3 and concerning directions of line segments in the boundary of K. In GALE's first paper he uses the Borsuk-Ulam mapping theorem (involving antipodal points) to prove that if a convex body of width w' in EM is obtained from one of width w by means of a homeomorphism which decreases distances, then w' w. COXETER proposes an exact upper bound for the number of equal nonoverlapping spheres in EM that can touch another of the same size. The dif. Can a rigid ficulty of this problem is indicated by the following quotation: material sphere be brought into contact with 13 other such spheres of the same size? Gregory said 'Yes' and Newton said 'No', but 180 years were to elapse before a conclusive answer was given." His historical survey of the problem in EM extends from a paper by Kepler in 1611 to the latest published works. The problem is treated as the case çS = irI6 of the problem of packing (n — 2)spheres of angular radius # on an (n — 1)-sphere, and the proposed upper bound is attained when the (n — 2)-spheres are inscribed in the cells of a regular polytope {p, 3, - -•, 3}. Though the bound is not fully established, much supporting evidence is given. Some related material is also discussed, such as the growth of the number of spheres as n — oo and the known results for other values of PORITSKY treats a system of linear inequalities of the form x,fi(O) + -•- +
g(O), where g and the f1's are real analytic functions of the real variable 0 ranging over a bounded or unbounded interval 1. He studies the convex region consisting of all points x = (x1, .. ., EM which satisfy the given system of inequalities (for all 0 1), and is especially concerned with describing the region's boundary in terms of the envelope curve C and its tangent and osculating flats of various dimensions, where C is the set of all = points x such that for some 0 I,. for 0 j n — I ((1) indicating the jth derivative). DVORETZKY reviews his earlier results on near-sphericity in EM, one of which asserts that for each e £ JO, 1[ and each positive integer k there exists N(k, e) such that every convex body of dimension N(k, admits a b-dimen-
INTRODUCTION
xi
to within e. He derives new corollaries, including some on orthogonal projections, and discusses some open problems. Two papers treat the facial structure of convex polyhedra. GALE's second paper is concerned with cyclic polytopes in R these being convex polyhedra which are combinatorially equivalent to the convex hull of an n-pointed subset of the moment curve {(t, t', . . ., ta"): t R}. They have the remarkable property of being rn-neighborly in the sense that each m vertices determine a face. He computes the number of (2m — 1)-dimensional faces of such a for convex polytope and this is conjectured to be the maximum which have n vertices. Certain neighborly polytopes are polyhedra in sional section which is
proved to be cyclic, and regular cyclic polytopes are constructed in GRUNBAIJM AND MOTZKIN call an abstract graph k-polyhedral
it is isomorphic with the graph formed by the edges and vertices a k-dimensional convex polyhedron. They prove that each k-polyhedral graph contains as subgraph a refinement of Ck+i, the complete graph with k + 1 nodes. As Gale's result shows, the graph is i-polyhedral whenever 4 j k; however, this and other sorts of ambiguity are excluded for graphs which are 2polyhedral or 3-polyhedral. VALENTINE deals mainly with known results on the intersection properties of convex sets. He obtains refinements and new proofs for many of these, his aim being to show what can be accomplished by systematic exploitation of dual cones. His viewpoint is well expressed by the following quotation:
"•-• since it is a rare coincidence for the proofs of a theorem and its dual to be of equal difficulty, there is a double reason to investigate the dual. One may gain either a simpler proof or a less obvious theorem." Five of the papers are expository surveys of a sort which should be valuable in any field, especially in the field of convexity where so many results
have been rediscovered so many times and where there are so many elementary unsolved problems. Though including few proofs or none at all, they give rather complete descriptions of known results and existing literature in their respective areas. Some of them include new results as well, and most of them discuss many unsolved problems. Since the papers are themselves summaries, it is hardly feasible to summarize them here, but it may be helpful to list their section headings. GRUN BAUM, Borsuk's problem and related questions — reductions of the problem; partial solutions; universal covers; other results on partitions; coverings by translates; finite sets; related problems. GRUNBAUM, Measures of symmetry for convex sets — distance-functions
for spaces of convex sets; invariant points and sets; a property of some measures of symmetry; general methods for geometric definitions of measures of symmetry; known results on special measures of symmetry; some extremal problems which possibly lead to measures of symmetry; an interesting functional; some generalizations. DANZER, GRUNBAUM AND KLEE, Helly's theorem and its relatives — proofs
of Helly's theorem; applications of Helly's theorem; the theorems of Carathéodory and Radon; generalizations of Helly's theorems; common transversals; some covering problems; intersection theorems for special families;
xii
INTRODUCTION
other intersection theorems; generalized convexity. (The last section makes little contact with the others. It contains a rather complete survey of existing generalizations of the notion of convex set.) KLEE, Infinite-dimensional intersection theorems — intersection theorems for infinite families (also in Re); intersection theorems involving the weak topology; intersection properties of metric cells. CUDIA, Rotundity — rotundity and smoothness properties; comparison of properties; product spaces, quotient spaces, and subspaces; duality; geometry and reflexivity. Like those of Cudia and Klee, the papers by BISHOP AND PHELPS and by PHELPS are concerned with the geometry of infinite-dimensional convex sets. The principal result of Bishop and Phelps is that if C is a closed convex
subset of a Banach space, then the support points of C are dense in the boundary of C. They show also that for each bounded closed convex subset C of a Banach space E, the members of the conjugate space E* which attain their maximum on C are dense in E* (norm topology). Several other interesting results are obtained by the same methods. The paper by Phelps treats some of the more technical points which arise when the space is not normable. In particular, he uses supporting cones to give a new proof of the existence of relative extreme points, where a convex cone K with vertex x is said to support the convex set C provided C n K = {x}. CORSON AND KLEE show that the topological classification problem for closed convex bodies in a normed linear space E can be reduced to that for E's unit cell and its closed linear subspaces of finite deficiency. For all dimensional spaces as well as for a wide variety of infinite-dimensional Banach
spaces, the problem is solved by proving that all closed convex bodies in E are homeomorphic with E itself. The main tool is the fact that certain spaces are homeomorphic with their positive cones. Also obtained are some results on uniformly continuous transformations of convex sets. The remaining papers are not so directly concerned with convex sets as such, though in each case some sort of convexity is essential either in the paper itself or for its motivation. Both Karlin and Davis deal with convex
For real intervals X and 1, KARLIN considers the functional transformation T carrying a real function f on Y into the function g = Tf on X given by the formula gx = K(x, the kernel K being a bounded measurable function on the rectangle X x Y. He is especcially interfunctions.
ested in conditions on K which insure that g is convex whenever! is bounded and convex; a similar problem for monotone functions is also considered. The conditions obtained involve the total positivity or sign-regularity of K, where K is said to be sign-regular of order r provided there exists a sequence numbers each either +1 or —1, such that whenever x1 <x, < --< - - < y11,, x, X, Yi Y, and 1 m r, then e.1K(x1, - - -, x1; yj, -. -, y.) 0, Ya where the expression K( ) is the determinant of the matrix which has
y) in the ith row and the jtb column; K is totally positive of order r provided this condition holds with all the e1's equal to +1. Inter-relation-
xiii
INTRODUCTION
ships among given.
various classes of kernels are studied, and many examples are
In his second paper, DAVIS studies various classes of real-valued convex functions (of one or several real variables) where for each class the defining of n x n (real) symmetric matrices. For cx condition involves the class
ample, if I
is
a function of one real variable and the matrix A
has
it is customary to write J(A) = In this way I can be regarded as a function on fin to fin. The function , (1 — .01(A) + Af(B) lot is called matrix-convex provided f((1 — A)A + AR) spectral representation A =
all A e LO, 11 and A, Be H,,, where the ordering is that induced in H,, by agree-
ing that a member of H, is non-negative if and only if it is positive semidefmite. The matrix-conyex functions form a proper subclass of the ordinary convex functions and are closely related to the matrix-monotone functions of Loewner. The paper is devoted to an exposition of Loewner's theory along with related ideas for several variables due to Korányi, Sherman, and Davis himself.
In addition to the paper of Poritsky mentioned earlier, two other papers are included here because of the close connections between convex sets and linear inequalities. BELLMAN AND FAN study systems of linear inequalities in which the variables are Hermitian matrices and the Ordering is defined as in the paper of Davis just mentioned. They find consistency conditions for various systems of inequalities, the conditions being quite analogous to those in the classical situation except that in each case the consistency of an auxiliary
system must be assumed. Also included are several interesting examples, as well as results on the minimum and maximum of the traces of certain matrices related to the systems in question. HOFFMAN supplies a unified approach to some linear programming problems which are amenable to "obvious" solutions. His guide is the observation by
Monge that if unit quantities are to be transported from points X and Y to points. Z and W (not necessarily respectively) so as to minimize the total distance traveled, then the two routes cannot intersect. He defines a Monge sequence to be an ordering of the set -((i,j): 1 i m, 1 j n) and introduces the notion of such a sequence being consonant with a given m x n matrix.
An algorithm is given whereby a solution for the transportation problem associated with a given matrix can be derived from a Monge sequence consonant with the matrix. The warehouse problem of Cahn is transformed into one to which this algorithm is applied and many other problems are mentioned to which the same idea is applicable.
The many new notions in MOTZKIN's paper are treated in 79 theorems distributed among 50 sections. The paper is concisely written and can hardly be summarized here, but we shall describe its basic idea. Let R be a (not necessarily commutative) ring with unit 1 and let V be a left module over R. Let R be the set of all finite sequences A = (A1, - - -, of members of R. When S c V, the vector A is said to be an endovector of S, or S is said to be endo-A, provided S includes the point A,s, for every choice of S
S
A
4.
Since
the family of all endo-A sets in V is intersectional, the A-hull of S is defined
INTRODUCTION
as the smallest endo-A set which contains S. The set A is said to be complete provided for some S, A is the set of all endovectors of S. These and related notions are studied in some detail, where of course the most important cases
are those in which R is the real field and the condition (As, ..-,
A
is
= 1; (iii) A, 0; (tv) equivalent to one of the following: (i) A E R; (ii) — 1 and A1 0. The corresponding endo-1 sets are the linear subspaces (0-flats), the affine subspaces (flats), the positive cones (convex cones with vertex 0), and the convex sets. MOTZKJN AND STRAUS are concerned with representing the points of a set as linear combinations of boundary points. Their principal result asserts a, for 1 that if a, + n, then for very + = 1 and general sets S it is true that each point of S can be represented in the form p= a,rx1 for points x of the outer boundary of S. PTAK presents a unified treatment of several important results on weak compactness, all of which are shown to follow from a combinatorial lemma which gives conditions for the existence of certain convex means. For an infinite set S, let M(S) denote the set of all functions A on S to [0, 001 for which the set N(A) is finite and A(s) = 1, where N(A) = {s S: A(s) > 0). Let be a of subsets of S, and for e > 0 and Hc S let M(H, e M(S) such that N(2) c H and Zsi,ewA(w) < e for all I
W€ (1) M(H,
j
The lemma asserts the equivalence of the following two conaitions: e) = 0 for some infinite H cS and some > 0; (2) there exists a
sequence (se) of distinct points of S and a sequence (W,,) of members of such that {sj, . ., c for all n. With the aid of this lemma he proves that if A is a subset of a complete convex space E and A satisfies a certain double limit condition, then the closed convex hull of A is weakly compact. This includes the well-known theorems of Krein and Eberlein on weak compactness. The same lemma is employed to yield an extensive series of results on weak convergence and weak compactness in locally convex spaces and especially in spaces of continuous functions.
SCHAEFER is concerned with spectral properties in an ordered locally convex algebra A, where this is a locally convex algebra (usually over the complex field) with unit e and with an associated positive cone K 3 such that K is closed, proper, includes the product of any commuting pair of its elements, and is normal in the sense that there is a family of pseudonorms p on E which generate the topology and are such that p(x + y) p(x) for all x, y e K. The principal motivating example of such an A is the algebra of all continuous endomorphisms of a Hubert space, where K is the cone of positive Hermitian operators and the topology is that of either bounded or pointwise convergence. (There are other important examples also.) The paper contains much interesting material on such algebras A, its principal results showing that the spectral behavior of certain members of K is quite analogous to that in the finite-dimensional case. In particular, the members of K whose spectrum is bounded have spectral behavior like that of positive matrices, while those in the unit interval of K (i.e., those a A for which 0 a e diagonal positive matrices in the classical case) behave spectrally like positive Hermitian operators.
xv
INTRODUCTION
FAN's paper is motivated by the Krein-Milman extreme point theorem. He establishes a general lemma which is purely set-theoretical in character, involving neither topological nor vector space concepts, from which the Krein-
(Another lemma, in a sense dual to the first, is shown to imply theorems on filters due to Waliman and Stone.) He then considers a set *P of real-valued functions on a set S, calling a set X C S convex provided X is an intersection of sets of the form {x E S:f(x) a). Since the family of 0-convex sets is intersectional, the 0-hull can be defined in the natural way. The notion of 0-betweenness is defined for points of S and in terms of this the 0-extreme points of subsets of X are defined. These notions appear in several theorems which generalize known results on extreme points and are related to the abstract minimum principal of Bauer. HAMMER's second paper is motivated by his notion of a semispace at a point p in a linear space L, this being a maximal convex subset of L '— {p). He reviews some of the known results on semispaces, including their connection with extreme points and the fact that the semispaces form a minimal intersection base for the convex subsets of L. He then describes his system Milman theorem follows.
of extended topology which arose from an attempt to consider certain processes
and concepts associated with convexity (and especially with semispaces) as topological in character. Many new notions are introduced, complications arising mainly from the fact that in place of the usual topological closure operation he considers an arbitrary expansive function g—i.e., one associating with each set some superset thereof. After discussing the extended topology, he interprets the various notions in terms of convexity, where yX
is the union of X with all the line segments determined by points of X. Several unsolved problems are mentioned. V.K.
ON SYSTEMS OF LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES BY
RICHARD BELLMAN AND KY FAN' 1.
IntroductIon. This paper is concerned with systems of linear inequalities,
in which the variables are Hermitian matrices. The inequalities between Hermitian matrices are to be understood in the sense of positive semidefiniteness or positive definiteness. More precisely, for two Hermitian matrices H and K of same order, we write H K to signify that H — K is positive Similarly, the strict inequality H> K means that H — K is positive definite. Consider the system (2) of linear inequalities, where are arbitrary square complex matrices, B and C1 are Hermitian matrices (all of same order), and c is a real Theorem 1 gives a necessary and sufficient condition for the system (2) LO be consistent, i.e., for the existence of matrices X, satisfying (2). Consistency conditions for more special systems (10), (14), (18) and (22) are also explicitly stated. Then we derive two theorems on minimum and maximum (Theorems 2,3). Theorem 2 asserts that the minimum of the trace of CX,, when {Xj) varies over all solutions of the system (14), is equal to the maximum of the trace of when {.Y1} varies over all solutions of the system (18), provided that the systems (14'), (18') of strict inequalities are consistent.
results are analogous to the well-known theorems on systems of linear inequalities in real variables (see [2; 3]). However, for the case of Hermitian matrix variables, each of our theorems requires an additional hypoth-
esis which is not needed in the case of real variables. Thus, in Theorem 1 we assume that there exist positive definite 1-lermitian matrices
satisfy-
ing (1); in Theorem 2 we assume that the systems (14'), (18') of strict inequalities of the systems (14), (18)) are consistent. These hypotheses are indeed essential (see Examples 2, 3).
The systems studied in \his paper are quite natural, especially in the case = n = 1. For instance, when m = n = 1, system
YA+AtY=-C, where A is aft arbitrary square complex matrix and C, Y are Hermitian. This differs
from the familiar system
Y>0,
YA+A*Y=_l
(I being the identity matrix), which arises in stability problems of differential equations. It a classical theorem of Lyapunov (see [1, Chapter 13; 4, "The twork of the second author was supported by the U. S. Atomic Energy Commission at Argonne National Laboratory.
RICHARD BELLMAN AND KY FAN
Chapter XV, § 5D that there exists a positive definite Hermitian matrix Y satisfying YA + At Y = —I if and only if all eigenvalues of A have negative real parts. All matrices considered here are square matrices with complex elements. Since the matrices considered in a theorem (except in the proof of Theorem 1) are always of same order, the order will often not be mentioned. Through out the paper, A,j are arbitrary square complex matrices which are not are Hermitian matrices. As usual, necessarily Hermitian, B, C,, X1 and
the adjoint of a matrix A is denoted by At, the trace of A is denoted by tr A. We repeat again that, for two Hermitian matrices H, K of same order, H K means that H — K is positive semi-definite, and H> K means that H — K is positive definite. 2. Exletence theorem. We begin with a lemma which will be needed in the proof of Theorem 1. LEMMA. Let H1, H, be p Hermitian matrices of same order. Let denote
the convex cone in the Euclidean fr-sfrace
formed by all points with
(tr ff2, tr H2Z, ..., tr H,Z), when Z varies over all positive semidefinite Hermitian matrices (of the same order as the He's). If there exist real is a closed numbers c1, c2, •, c, such that cd-ft is positive definite, then coordinates
set in PROOF.
Let Z1, Z2, . . ., Z1,--- be a sequence of positive semi-definite Hermitian
tr ]natrices (of the same order as the Hi's) and let We want to prove that (t1, . ., t,)e Let H0 =
= tt (1 k f). CJ-4 be positive
definite, and let a be a positive number such that al H., where I denotes so there exists a the identity matrix. We have trH,Z1 = positive number b such that a- trZ1 b for all i = 1,2,3, . - - Then 0 -c trZ b/a for all i. Since Z1 is positive semi—definite, trZr (bfa)2, which implies that the elements of are bounded. There exists a sul)sequence {Z} such that lim,... Z exists. Then Z is a positive semidefinite Hermitian matrix and tr HkZ = lim,.... tr = tt (1 k P). Hence -
(li,
--,l,)e
EXAMPLE 1.
Let
H2=(° The set
in
.
formed by all points with coordinates (tr H1Z, tr 1hZ), when
Z varies over all positive semi-definite Hermitian matrices of order 2, is not closed. In fact, consists of the origin (0,0) and all points (x, y) with x > 0 and —00 < +00. THEOREM 1. Let A1, be arbitrary matrices, B1, C .be Hermitian matrices (all of same order), and let c be a real number. SupPose that there exist positive definite Hermitian matrices (I)
satisfying
LINEAR INEQUALITiES IN HERMITIAN MATRIX VARIABLES
Then the system
trtCjXj
c
is consistent, i.e., solvable for Hermitian matrices X,, if and only if, for any ,n Positive semi-definite Hermitian matrices the relations
and any non-negative number d,
+ dC1 =0
+
(1
j n)
imply
0.
+ dc
PROOF. Necessity. Assume that X,, X2, •, X,, are Hermitian matrices satisfying (2). For any m positive semi-definite Hermitian matrices A and any non-negative number d, we have •
trZD1B, + dc,
j=1
which can be written 1=1
i=i
+ A1D1) + dC1}X1 i=1
+ dc.
Hence relations (3) imply (4). Sufficiency. Let denote the real vector space of all Hermitian matrices (of the same order as the and let denote the one-dimensional vector space (i.e., the real line). Let . be the direct sum with m summands. In the vector space a vector a = (S1, S2, - -, S,,,, s) is determined by- m Hermitian matrices S. and a real number s. In an inner product is defined in a natural way as follows. For two vectors a = (S1, .. •, S,,,, s) and r = (T1, -. -, T,,., t) in the inner product (a, i-) of a, r is
(a, r)
= trES.T. + st.
We can therefore speak of orthogonality in Let denote the linear subspace of the form + where
...,
(A,,,1X3
formed by au vectors of
+
tr±C1X1)1
Let
=
Dk2,
.
. .,
dk)
(1
k
p)
RICHARD BELLMAN AND KY FAN
be p vectors which span the orthogonal complement E dk e (1 k 1'). (B1, B2, ••, B., c) and bk = —(ôk, Let of ..9' in •', S,., s) S2, is formed by all vectors c = (S1, Then the linear variety 5' — a) (1 5 k p), i.e., satisfying (0k, = bk in (where
(lSkSP). such Let i? denote the set of all vectors (Z1, Z2, ••, Z1., z) in that each Z, is positive semi-definite and z 0. In the Euclidean p-space let
denote the convex cone formed by all points with coordinates of
the form
+ d1z, ", when (Z1, Z2,
•, Z,,1, z) varies in
+
For k = 1,2,
'Ik = diag. {Dkl, Dkl,
• •,
let
Dk.,, dk}
dk are its successive denote the Hermitian matrix such that Dk2, diagonal blocks and all elements outside these blocks are zero. It is clear that Q coincides with the set in formed by all points with coordinates of the form (trH1Z, trH2Z, ..., trH,,Z), when Z varies over all positive semidefinite Hermitian matrices (of the same order as the Ilk's). According to the above lemma, in order to show that Ic" is closed in it suffices to find real numbers c1, c2, ., such that ckHk is positive definite. Now, by hypothesis, there exist m positive definite Hermitian matrices Y1 satisfying (1). So we have
+ =
tr±
+ tr±C1X, +
+ c1}x1
0
for any X, e Thus in the vector 7j = (}'1, 1's, - --, 1'1., 1) is orthogonal to the linear subspace .5'. Therefore is a linear combination of 8,, -, 8,,. We can find real numbers c,, c,, -, C,, such that 1', = c,Dk1 - -
(1 i S m) and I =
Ckd,.
Then
Y,,--,Y1.,1}, i.e., }'1, }',, ..., Y.,, 1 are the successive diagonal blocks in the matrix c,,J-4, and all elements outside these blocks are zero. Hence ckHk is positive
definite and is closed in Assume now that system (2) is inconsistent, which means in view of the characterization (5) of .5' — and our definition of i", (6) amounts to saying that in the point (b1, b,, .. -, b,) is not contained in the
LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES
closed convex cone
which separates there is a hyperplane in strictly. In other words, we can find p real
Hence
and the point (b1, b2,
5
•, b,,)
a, such that
numbers
\
I
p
p
> k=1
for all (Z1, Z2, •••,
As
z)
=
=
dkc
—
we have
+ c)} >0
÷ B,)
for all (Z,,Z,, •••,Z,,,,z)E.y'. Let p
D1
= ZakDk1(1
I
d=
m),
p
k=t
Then
tr for all (Z1, Z,,
+ d(z + c) > 0
D1(Z, +
This implies
•, Z,,,, z)
(7)
and
trZ D1B1 + dc > 0.
(8)
On the other hand, since the vectors
8,
have ;,_1
i=1
we
trZC1X1 =0 1-4
+
,=I
for 1 k
+
Dk,(A1JX, +
are orthogonal to
+ dkCI}XJ =
p and for any arbitrary X, e +
0
Hence
+ dkC, =
0
(1
k p, 1 j
and consequently
Combining this with (7), (8), we infer that the condition stated in the theorem
RICHARD BELLMAN AND KY FAN
6
is not satisfied. Thus the sufficiency of the condition is proved. 3.
ApplIcation to special syatems. We consider some special cases of
Theorem 1. First, in the case C = to the following corollary. COROLLARY 1.
0
(1 j
n) and c =
0,
Theorem 1 reduces
Suppose thai there exist Positive definite Hermilian matrices 1'
satisfying
Then the system
(1 i m)
+
is consistent (i.e., solvable for Hermitian matrices X,) if and only if, for any the relations
m Positive semi-definite Hermitian matrices
+
=0
(1 j
n)
0. As a special case of Corollary 1, we have 2.
Suppose that there exist Positive definite Hermitian matrices Y
satisfying
+
0,
we derive from (27)
(1 (1
n), I m)
RICHARD BELLMAN AND KY FAN
10
d.trZKiB, tr
tr±C1M,,
+
In case d = 0, we use the fact that (14), (18) are consistent. By Corollaries 2, 3 (see also Remark after Corollary 4), relations (27) (with d = 0) 0, and therefore again (26). This M1C1 0 and tr imply tr completes the proof. For the systems (10) and (22), we have THEoREM 3. Suppose that the systems whence (26).
+
i
(1
m)
(22')
Cix,, when {X1} varies over alt are consistent. Then the minimum of tr when solutions of system (10), is equal to the maximum of tr varies over all solutions of system (22). PROOF.
As
in the proof of Theorem 2, it amounts to show that the system
tr(tBiYi — tc1x3) 0 •is consistent. The consistency of the systems (10') and (22') means precisely that the hypothesis concerning (1) in Theorem 1 is satisfied for the present system (28). According to Theorem 1, (28) is consistent if and only if, for Hermitian matrices any arbitrary Hermitian any positive matrices K' and any non.negative number d, the relations
(1
imply
tr(tJi.Bi —
0.
i
m)
LINEAR INEQUALITIES IN HERMITIAN MATRIX VARIABLES
This implication is easily verified (by using Corollaries 1, 4 for the case d = EXAMPLE 3.
0).
Let —
/0
0\
A Hermitian matrix X =
\Z
B — 10
)'/
0\
c
—
(0
1
satisfies
0, AX+ XA* B if and only if x = 0, y 0 and z = 0. Hence, for every solution X of (29), we satisfies have tr CX = 0. A Hermitian matrix V = \w VI
0, VA + A5Y
C
0. So we have tr BY = —1 for 1, wI: u and w + u' every solution Y of (30). This example shows that the hypothesis concerning (14'), (18') in Theorem 2 is essential.
if and only if v =
REFERENCES
1. R. Bellman, Introduction to matrix anaLysis, New York, 1960. 2. K. Fan, On systems of Linear inequaLities, Linear inequalities and related systems, pp. 99-156, Ann. of Math. Studies, No. 38, Princeton Univ. Press, Princeton, N. J., 1956. S. D. Gale, H. W. Kuhn and A. W. Tucker, Linear programming and the theory of
games, Activity analysis of production and allocation, pp. 317-329, Wiley, New York, 1951.
4. F. R. Gantmacher, The theory of matrices, Vol. II, translated by K. A. Hirsch, Chelsea, New York, 1959. RAND CORPORATION AND
NoamwzsTKEN UNIVERSITY
MINIMUM AREA OF A SET OF CONSTANT WIDTH BY
A. S. BESICOVITCH
The area of a set of constant width d is a minimum when the set is a Reuleaux triangle. Like the proofs of Lebesgue [1) and Eggleston [2], the
present one depends on the fact that about any set of constant width d a regular hexagon of the same width can be circumscribed. Let A1, A., .. ., A, be the vertices of the hexagon, and B1, B.,
., B, points of contact of the hexagon
A3
P
A1
FIGURE 1
with the set of constant width where B1 is on side A1A. a.s.o. A point M of
the boundary of the set is defined by the angle 8 between A1A. and the tangent PM at M where P is on A1A.. Let R(8) be the radius of curvature at M so that R(8) + R(8 + ,v) = d, and let MN be the perpendicular to A1A.. We have, for 8 = MN =
sin 8d6, 2
(the sign
I
I
MP1dO
is for area), so that 13
MP = MN/sin
A. S. BESICOVITCH
14
(1)
I
B1B2A,I +
i = —' 2
We have
it,
.
d
nt.R(O) sin OdO
I j, sin ciu.
+
tjo
.1
it,
R(O
+
sin OdO
'I,
R(O ÷
MO] =
sin OdO}
where the sign of equality takes place only if one of the two integrals on the left side is zero; that is, the sum (1) takes its maximum value only when (pr, + ir/3). Similar conR(8) is zero throughout one of the intervals (0, clusions hold for the sums of the other pairs of areas and thus the total area of the part of the hexagon that is outside the set reaches its maximum only when the set is a Reuleaux triangle. REFERENCES
1. T. Bonnesen and W. Fenchel, Th.ori. der konvexen Korper, J. Springer, Berlin, 1934. p. 132.
2. H. G. Eggleston, A proof of Bia.chke's theorem on the Reuieanx triangle, Quart. J. Math. Oxford Ser. (2) 3 (1952), 296-297. UNIVERSITY OF PENNSYLVANLA
ON SEMICIRCLES INSCRIBED INTO SETS OF CONSTANT WIDTH' BY
A. S. BESICOVITCH
V. Klee has set the problem: Can a semicircle of diameter 21 be inscribed into any set 1' of constant width
A partial solution of the problem is given by the THEOREM. Into every simple set 1' of constant width 21 can be inscribed three semicircles of diameter 21.
We call a set of constant width simple if its boundary has a curvature at every point and if the set of centers of curvature is a triangle M'M"M" formed by three inwards convex arcs. The theorem holds under more general conditions and is likely to be true
in the general case. We shall call the boundary of I' also 1'. There is a unique point Ma on M"M" such that' =
—
There are similar points
and
—
on
the other sides of the triangle. Every
normal of 1' is a "double normal," that is it is the normal at each end; it is of length 2!, and it touches the triangle M'M"M" at the point that is the common center of curvature of I' at the end points of the normal. There are these relations between radii of curvature Md'2, MaP4 at different points of 1': + M1P, = —'M"1W2 + Ii1,P,
,
M1P1
+ 1$/laPa
M4Pa = MuNa = 1
similar ones. The other end of the normal at P. P',
and
N,N',...,N,,...
P1,
is
denoted
sothat
NP=N'P'=...
=21.
The end-points above P0N4 are denoted by P, and those below by N. The theorem will be proved by proving that the top half of F bounded from below by NaPa contains the top semicircle on the diameter for which we have
to show that for any point P on the top half of the boundary of r, MaP 1. The line MIIM' meets at most one of the sides M'M", M'M" in a second I am indebted to Dr. Danzer for criticism and suggestions to my first version of this note. 2 The arc on the diagram with end-points A and B, and its length are denoted '—AB. The segment joining points A and B, and its length, are denoted AB, whether the segment itself is on the diagram or not. 15
A. S. BESICOVITCH
P2
/
It,
FIGURE 1
point different from M', so that a line through Ma may touch at most one of the sides M'M", M'M" at a point different from W. Suppose it meets M'M" The proof for this case will also include the case when MaM' does in not meet M'M", M'M" in any point different from M'. Consider the normals of 1' touching M'M"M" at various points M1, M2, M1€
We have =
+ M.P,.
On the other hand M1P1 <M1Ma + MaP1
By (1) and (2) MaP1 > MaPa
=1.
The same is true for M1 e We have + M2P2 = M"Ma + MaPa.
ON SEMICIRCLES INSCRIBED INTO SETS OF CONSTANT WIDTH
On the other hand + M0P. By (3) and (4) M0P2 > MaP6 = 1. The same is true for M2 E M"M'. ÷ M2P2
P3M,, + M0M" > P3M3 + M3Q + '—QM".
By (5), (6), (7)
P3Ma +
>
+ M0N0.
Hence
P3M,, > M0N0 = 1,
which completes the proof that the top semicircle on the diameter N4P0 is included in I'. We have similar results with respect to the semicircles corresponding to the points Mb and M4, which prove the theorem. REMARK. Our theorem has been proved by showing that a simple set always includes one of the semicircies constructed on each normal that is bisected by the center of curvature at end-points of the normal. The question arises whether this is also true in the general case. The answer in the negative is given by the Figure 2. Let ABCDEA (Figure 2) be a pentagon symmetric
with respect to the bisector of the angle CDE and suppose that it is the evolute of a set I' of constant width d. If the sides of the pentagon are linear segments then of course only vertices are centers of curvature. We
shall assume that they are smooth curves of convexity, the direction of convexity being obvious (e.g., AB has its convexity directed upwards). We shall treat the pentagon as linear, but it is obvious that the argument is valid for the case of sides of small convexity. Because of the symmetry of the pentagon the midpoint of AB bisects the normal touching the pentagon at Similarly the points 0,, 0,, 0, such that A01 = A0,, EO, = E0,, DO, = DO4, CO4 = CO (BO, = DO1 because of symmetry) bisect the normals touching other
A. S.
18
B
B
A
FIGURE 2
sides of the pentagon. The endpoint P of the normal rotating about A in Consequently the arc of the angle BAE is distant more than d/2 from the semi-circle center and radius d12 in the angle BAE is inside I'. Hence the opposite arc of the circle is outside I'. Thus only the upper semi-circle may be included in 1'. On the other hand, when the normal rotates about 0 its midpoint describes the circular arc 04003,0 being below and therefore
the top end of the normal through D and 0 is within less than d/2 from that is inside the semicircle. The semi-circle is not included in 1'. OF PENNSYLVANIA
A CAGE TO HOLD A UNIT.SPHERE BY
A. S. BESICOVITCH
Problems on a net to hold a unit-sphere and a box (a polyhedral surface) to hold a unit-sphere have been considered [1; 2]. Coxeter has set a problem on a cage (a frame formed by edges of a convex polyhedron) to hold a unitsphere [3]. The problem is: Find a cage of minimum sum of edges, to hold a unitsphere and not permitting it to slide out. Coxeter expressed a conjecture
that it iS a right triangular prism all of whose edges are equal to i/s, so that = 15.59. This conjecture is false, as follows the sum of all the edges is from the THEOREM. 71w greatest lower bound of the sum of edges of a cage to hold a unit-sphere is
r
= 8ir/3 +
11.88.
The theorem will be proved by showing that for any £ > 0 a cage can be defined for which the total sum of edges is less than r + E. Given a circle of radius r, take the tangent to it at the point M and on the tangent take a segment AB of length 2'1/3r/3 with mid-point at M. From the points A and B draw the second tangents, touching the circle at the points E and C, respectively. We have AE = BC = './3r/3. The length of the curve consisting of the polygonal line EABC and of the arc CDE is equal to + 4n-/3)r. and Divide now the arc CDE into n equal arcs by points P1. P. . .., circumscribe the polygonal line about CDE touching CDE at the points C, P1, P2, -
-, E. Denote its vertices Q1, The polygon ABQIQI -• . , (M, circumscribing the given circle. When n lends to oo the perimeter of the (M, n)-Polygon tends to
will be called + 4ir/3)r.
Take now a great circle of the unit-sphere and a point M on it. Through
the tangent at M draw two planes forming angles a and —a with the great circle, with small a. The planes meet the sphere in two circles of radius r = cos a. Let (M, n)-polygons circumscribing these circles be ... Q1'A. These polygons, together with the -• Q1A and form a polyhedron. The boundaries of the two poly"joints" QQ', gons touch the sphere, while for a fixed n and for a sufficiently small all the joints are outside the sphere. Thus the edges of the polyhedron form a cage and the sphere cannot slide out of it. We shall now fix n so that the perimeter of each of polygons be less than . -. + 4ir/3 + e/3, after which we shall take a small (4V3/3 + 4r/3)r + e/3 < enough for all the joints to be outside the sphere and for their total length to be less than e/3. With these conditions, the total length of the edges of the cage is + e, which proves the theorem. • -
1.9
A. S. BESICOVITCH
20
The question whether r is the greatest lower bound remains open. For finding the lower bound, it is sufficient to consider cages of total length of edges 12. That means that at least one face circumscribes a circle of radius = = .866 (call it a base). If there were always more than one base, then it would be easy to show that the smallest perimeters belong to the cages of the kind we have described. If this is not
so then, if the base is a triangle, its perimeter is so large that it is easy to show that the perimeter of the whole cage is > r. If on the other hand, the perimeter of the base is not so large than the base is a polygon of 4,5, 6, sides and, because many sides of the cage start from the base, it may be possible to show that in this case the remaining part of the• perimeter is too large; Remember that the perimeter of the circle of intersection of the base = with the sphere is = 5.45 and if the base is a triangle then its
perimeter is >9, if it is quadrilateral then perimeter is >7, pentagon >6.5, hexagon >6.
1. A. S. Besicovitch, A net to hold a apheve, Math. Gaz. 41 (1957), 106-107. 2. A. 5. Besicovitch and H. G. Eggleston, The total length of the edges of a poiyh.Sron, Quart. J. Math. Oxford Ser. (2) 8 (1957), 172-190. 3. H. S. M. Coxeter, review of 2 in MR 20 $1950. UNwsasn'y OF PENNSYLVANIA
ON SINGULAR POINTS OF CONVEX SURFACES BY
A. S. BESICOVITCH
It is well known that the set of singular points on a convex surface in 3space is of a-finite linear measure. For a recent proof and for references to others, see R. D. Anderson and V. L. Klee, Jr. El]. In the present note I give an alternative proof. We shall consider ordered pairs of semi-planes (P, Q), with a common edge PQ, and not forming one plane. The convex hull of (P, Q) will be called the interior of (P. Q). Thus the interior includes the semi-planes. The direction of a semi-plane is defined by the direction of a vector perpendicular to the edge of the semi-plane, with the origin on the edge and the end point on the semi-plane. The angle between two semi-planes either belonging to the same pair or to two different ones is the angle between the two vectors. We take The angle between semi-planes of the same it always to be and pair is always <sr. The distance between two pairs (P. Q) and (P', Q'), A{(P, Q), (P', Q')), is defined by the larger of the angles between the semi-
p'anes P and P' and Q and Q'. Thus 4 is a metric. The set of all pairs (P', Q') for which 4{(P, Q), (P'. Q')} < 8, >0, is the 8-neighborhood of (P, Q). LEMMA 1.
Given a Pair (P, Q) with angle >0, and e > 0 there exists 8 > 0 such
that for any two pairs (P', Q') and (P", Q") of the 8-neighbourhood of (P, Q)
(1) The angle between the edges P'Q' and P"Q" is <e and (ii) if A' is a point of P'Q' interior to (P". Q") and A" a different Point of P"Q" interior to (P', Q') and different from A', A' is on P"Q" then I/ic angle between PQ and A'A" is <e.
(i)
From an arbitrary point M drop the perpendiculars MI" and Mq' on
the planes P' and Q' and MI" and Mq" on the planes P" and Q". The edge P'Q' is perpendicular to the plane M/."q' and the edge P"Q" to M/."q". The angle between the planes Mp'q' and Mp"q" is small for small 8 from which (i) follows.
(ii) Let a' be the angle between P' and Q' (Figure 1). Through the points A' and A" draw planes perpendicular to P'Q' and to P"Q". Their lines of
intersection with P' and Q', P" and Q", are
B'q4,
A"q'. B"A" is the segment of the edge P"Q" between the two planes. The segment B"A" meets one of the semi-planes, say P' in a point C between B" and A" unless P'Q' is parallel to P"Q" in which case P'Q' and P"Q" coincide and there is no problem. Drop the perpendicular CD on As in the right triangle CDB" the angle CB"D is nearly right and the angle CDB" is right, the segment CB" is large compared with B "D and also compared with B"A'. Hence A"B" is large compared with B"A'. Con21
A. S. BESIC0VITC}I
D
Pt
1
the angle B"A"A' is small, which proves (ii). REMARK. On the diagram the points A' and A" are strictly inside the angles respectively. It is clear that the conclusion holds also and when one of them or both are on the boundaries of respective angles. Let 0 < a0 < ir/2. Consider the set of all pairs (P. Q) for which the angle a satisfies the inequalities
p'B
a0
a
iv — a0
Obviously the Heine-Borel argument is applicable to the set and the set can
be represented as the sum of a finite number of neighbourhoods on each of which the Lemma 1 is satisfied for the same £. Hence the set of all pairs N1 of an enumerable set of such o < a < iv, can be represented as the sum neighbour/s oods.
ON SINGULAR POINTS OF CONVEX SURFACES
Let S be a convex surface. With every singular point A of S associate a pair (P, Q) of semi-planes of support at A, not forming one plane, and contain-
ing the whole of S in its interior. A lies on the edge PQ and is inside any pair corresponding to another singular point. Take of the above neighbourand let E1 be the set of all singular hoods a neighbourhood N1 of points of S to which correspond pairs belonging to N1. If A', A" are an arbitrary pair of points of E1 and (P', Q'), (P", Q") the corresponding pairs of N1 so that A' is on the edge P'Q' and in the interior of (P", Q") and A" is on the edge of P"Q" and in the interior (P', Q') then the angle between lie on a rectifiable curve and P,Q, and A'A" is <e. Hence all points of we arrive at the THEOREM. All singular Points of a convex surface lie on an enumerable set of rectifiable curves. COROLLARY.
The set of singular Points of a convex surface is of a-finite linear
measure. DEFINITION.
If there exists a nondegenerate right circular cone with vertex
at a point A of S. containing the whole S, then A is called a conical
The two-dimensional measure of the set of directions of the planes of support at A (that is of their perpendiculars) that do not contain any generator of the cone, is positive and at most at two points such planes
singular Point.
may be parallel. Hence THEOREM 2.
The set of conical singular points of a convex surface is at most
enumerable. REFERENCE
1. R. D. Anderson and V. L. KLee, Jr., Convex functions and upper semi-continuous functions. Duke Math. J. 19 (1952), 349-357. CAMBRIDGE UNIVERSITY
ON THE SET OF DIRECTIONS OF LINEAR SEGMENTS ON A CONVEX SURFACE DY
A. S. BESICOVITCH
The problem on the measure of the set of directions of linear segments on a convex surface has been set by V. Klee [11 and has been solved in 3-space by T. J. McMinn [21. In this note I give an alternative solution. The problem in 4-space remains still open. Let be a set of vectors in the 3-space. Take the set of unit vectors parallel to all vectors of with the origin at the same fixed point. Denote by E the set of end-points, and by the set of directions of vectors of Then the measure of is defined by the measure of E, so that = 4E, } = 4'E, and so on. (i) If is the set of all vectors in a plane then = 2,r. The same result holds if is the set of all vectors parallel to a fixed plane. (ii) Given a trapezoid with bases a and b and height h, if is the set of all vectors meeting each base then < 2(a + b)/h. (iii) If is a cone with the directrix of finite length then < If S is a closed convex set in the 3-space, we shall denote by CS) the of directions of all vectors lying on the surface S. THEOREM The set (S } is of a-finite linear measure. Given two closed convex curves C', C" in the two different parallel planes Z = Z', Z = Z". Denote by 2' the lateral surface of the convex hull of C' UC" (c.h. C' U C") and by 2" the set of all vectors lying in 2' which meet each of the curves C' and C". We shall first prove -
LEMMA.
AZ' is finite.
Consider a variable plane P touching C' and C" and having them on the same side. If p touches C' and C" each at a single point then the segment
joining them lies on I. If p touches one of the curves in a point and the other one in a segment then the triangle with the point and the segment as a vertex and a base respectively lies in 2'. Finally if p touches each C' and C" in a segment then the trapezoid with these segments as bases lies in 2'. We shall consider points of C' and C" as end points of variable vectors M'(t) and M"(t) with origins at the coordinate origin. We define these functions in the following way. Let the plane touch C' and C" at the points M,, MJ'. For 0 I AC' + AC", M'(t) and M"(t) satisfy the conditions AM0'M'(t) + = where M0'M'(t) and M0"M"(I) are the arcs of C' and C" respectively touched by the plane in varying from to p and the points M'(t) and M"(t) lie in the same plane p. It is easy to see that t defines the £
plane p always uniquely. The points M'(t) and M"(t) are also defined uniquely 24
DIRECTIONS OF LINEAR SEGMENTS ON A CONVEX SURFACE
25
unless p meets both C' and C" in segments, in which case we let first M'(t) describe the whole segment of C' keeping M"(I) stationary after which we keep M'(t) stationary and let M"(t) describe the segment of C". For every t, the segment M"(t)M'(t) lies in I, and if I has no trapezoid, these are all the segments of I meeting each curve C' and C". If there are trapezoids then, by (ii) the linear measure of the directions of all the segments with end points on bases of the trapezoids is less than 2(AC' + AC")/(Z" — Z'). It
remains to prove that the linear measure of directions of
all segments M'(t)M"(t) is finite. Denoting by m'(t), m"(t) horizontal components of M'(t)
and M"(t), we see that if we place the origins of the vectors M"(t) — M'(t) on the coordinate origin then the end points of the vectors will describe th curve m"(t) — m'(t) of finite length in the plane Z = Z" — Z'. By (iii), the linear measure of directions of all the segments ±(M"(t) — M'(t)) is finite, which completes the proof of the lemma. We pass now to the proof of the theorem. For all rational r denote by Cr the intersection of the surface of S with the plane z = r. Any vector on the surface of S either parallel to the plane z = 0 or meets for infinitely many values of r. Denote by the set of vectors parallel to z = 0 and by (r', r") the set of vectors of S meeting we have
{(r',r")J 2ir. Every vector of (r', r") belongs to the surface of for if it were inside it would be also inside S. Thus {(r', r")} C {c h 'C,'UC,") and by the lemma
By (i)
C
/t
U
A{(r', r")) < which proves the theorem. REFERENCES
1. V. L. Klee, Research problem No. 5, Bull. Amer. Math. Soc. 63 (1957), 419. 2. T. J. McMinn, On the tine segments of a convex surface in E5, Pacific J. Math. 10 (1960), 943-946. UNIVERSiTY OF PENNSYLVANIA
OF A CONVEX SET
THE SUPPORT BY
ERRETT BISHOP AND R. R. PHELPS The following well-known separation theorem is basic to the considerations of this paper. SEPARATION THEOREM -
Suppose
that A and B are convex subsets of a real
Hausdorff topological vector space E, and that the interior of B is nonempty and disjoint from A. Then A and B can be separated by a hyperplone, that is, there exists a continuous linear functional f 0 on E such that sup! (A) inf f(B). This theorem is a geometric version of the Hahn-Banach theorem. Its proof can be found in any of several texts, for instance in [3, p. 4171. An immediate corollary is the following support theorem.
If C is a convex subset of a real Hausdorff topological vector space E, if x is a point in the boundary of C, and tf the interior of C is nonempty, then there exists a hyperplane which supports C at x, that is, there exists a continuous linear functional f 0 on E such that f(x) = supf(C). We refer to such a functional f as a support functional of C, and x is called a support point. Note that if f is a support functional of C then every positive multiple of f is also a support functional of C. The assumption that C has interior points is a strong one, but some condition is indispensable to the validity of the support theorem. Indeed,
V. L. Klee has shown [6) that there exists a bounded closed convex subset of a dense subspace of a Hilbert space which has no support points. In the same paper Klee asked whether every bounded closed convex set C in a Banach
space has at least one support point. In this paper we answer Klee's question affirmatively: We show that the support points of C are actually dense in the boundary of C. This is shown to be true even if C is not bounded. Still assuming that E is a Banach space, we then prove (if C is bounded) that the support functionals of C are dense in the dual space E*. If C is not necessarily bounded, we show more generally that for each f in E* which is bounded on C and each £ > 0 there exists a support functional g of C with I If — gil <e. In fact it is shown that g can be chosen to strictly separate C from any bounded set X which is strictly separated from C by 1. We also show that every hyperplane which intersects the boundary of C contains a support point of C. Examples are given to show that these theorems fail in certain more general situations. The methods of this paper derive from a previous paper (1], in which a proof was indicated of the fact that the set of support functionals of a bounded closed convex set in a Banach space is dense in the dual space. By extending and simplifying the method of [1], we have been able to improve this 27
ERRETT BISHOP AND R. R. PHELPS
28
result and to obtain proofs of the related theorems mentioned above. Although all our theorems are stated and proved for spaces over the real field, they may easily be formulated and extended to spaces over the complex field by applying them to the underlying real space (obtained by restricting multiplication to real scalars) and to the real parts of complex linear function• als. These formulations are analogous to that given in [31 for the separation theorem.
Throughout this paper we will restrict our attention to normed spaces E and the convex sets under consideration will be assumed to be proper and nonempty. We write U for (x: lixil 1); if JEE*, then 11111 is defined to be supf(U). The proofs of the existence of support points and support functionals are based upon showing the existence of certain support cones for a given closed convex set. We will say that a subset K of E is a convex cone if K is a convex set and Ày e K whenever ye K and A 0. If X is a set containing the point x0, and if K is a convex cone such that K ÷ x0 is disjoint from X then we say that K + x0 supports x at x0. Suppose, now, that K has nonempty interior, that C is convex, and that K + x0 supports C at x0. Then, by the separation theorem, there exists a nontrivial g in E* such that supg(C) infg(K + xo). It is easily verified (since x0 is in both K + x0 and C) that supg(C) = g(x0) = infg(K + x0), i.e., g supports C at x0. Thus, to show the existence of support points and support functionals for C it suffices to find support cones of C which have interior points. The cones which we will use for this purpose are all of the following type: It! is an element of E* of norm one and jfk > 0, then K(f, k) kf(x)).
DEFINiTIoN.
{x: IIxfI
Clearly, K(f, k) is a closed convex cone; furthermore, if k> 1, then the interior of K(f, k) is nonempty. To see this, choose x in E such that II x = 1 and f(x) > Since f and the norm are continuous, and since x O. Ifz€X, then there exists a point x0 in X such that x0 e K(f, k) + z and K(f, k) + x0
supports X at x0.
PROOF. We partially order the set X by means of K; that is, x >- y means This, of course, is equivalent to saying that II x — y II kf(x — y). It is easily seen that if there exists a maximal element x0 in X, then K ÷ x0 supports X at x0. To obtain the conclusion that such an x0 exists for which x — yE K.
x, >- z, it suffices to apply Zorn's lemma to the set 2 of those x in X for which x >- z. Suppose, then, that W is a totally ordered subset of Z. The set {f(x): XE WJ is a bounded monotoriic net of real numbers (using W as
THE SUPPORT FUNCTIONAIS OF A CONVEX SET
our directed index set), hence it converges to its supremum. This implies
that it is a Cauchy net, and since x, y in W implies that
liz — y II
k[f(x) — f(y)J, say, we see that W itself is a Cauchy net in Z. Now, 2= X fl (K + z); since K is closed, Z must be complete and therefore W converges to an element of y in Z. By continuity of I and of the norm, it is simple to verify that y >- x for all x in W, i.e., W has an upper bound in Z. Zorn's lemma then applies and our lemma is proved.
The ideas we have developed so far enable us to prove the density of support points. THEOREM 1. If C is a closed convex subset of a Banach space E, then the support points of C are dense in the boundary of C. >0, choose y in E—C PROOF. If z is a point of the boundary of C and such that It y — z Ii 1 + 21€. If g is non-negative on K(f, k), then ill —gil PROOF.
h =g on f'(O) and IIhII =
ERRETT BISHOP AND R. R. PHELPS
30
in E such
lix II = 1 and 1(x) > kt(1 + 2/c), and 2/c. Then lIx±yli suppose that y in E is such that 1(y) = 0 and 0. This implies so x±yeK and henceg(x±y) 1 + 2/c < kf(x) = kf(x±y), Choose x
PROOF.
that
€/2 whenever f(y) = 0 1. Clearly, then, g(y) I that lg( y) I g(x) II x II c or II!— gil c. Choose z and 1, so by Lemma 2, either If + in E such that llzil = 1 and f(z) > c). Then zEK so g(z) Oand it follows that II! + gil (f + g)(z)> c, and our proof iS complete. We could now easily prove our density theorem; to include unbounded sets C, it would he formulated somewhat as follows: If f is bounded on C, then there is a support functional of C which is arbitrarily close to 1. With little care, however, we can do considerably more than this; the result is expressed as follows: If f strictly separates the set C from a bounded set X, then there is a support functional of C which is arbitrarily close to land which strictly separates X and C. More precisely: THEOREM 2. Suppose that C and X are subsets of a Banach space E, that C is closed and convex and that X is bounded and nonempty. If e > 0 and if f -
in E*, 11111 =
=
1,
is such that supf(C) < inf 1(X), then there exist g in
and x0 in C such that 111—gil s and g(x0) =supg(C) < infg(X). PRoof. Let r = supf(C), 8 = inf 1(X) and choose such that r < 8 < 8. Consider the neighborhood V of X defined by X + (8 — 8)U = V. This is a bounded set, and since inf f(U) = —1, we have inff( V) = inf 1(X) —(8— = Let a = 1 + 2/c and choose z in C such that r — f(z) < (2a1'(8 — r). Let M 1,
be larger than 2'(8—r) and sup{lly—zil:ye V} and let k=2aM(8—r)1. (Note that k > a> 1.) Choose, by Lemma 1, a point x0 in C such that K(f,k)+x0 supports Cat x0 and x0—zeK. We will show that VcK+xo. Indeed, if yE V, then fly — xoII iIY — zil + lIxo — all <M + fixo — zil M + kf(xo — a) M+ kEy —f(z)1< r)=2M< 2aM=h(8— kf(y — x0). By the separation theorem there exists g in E*, llgtl = 1, such that supg(C) = g(x0) inf g(K+ x0) inf g( V) infg(X) —(8— $) < infg(X). Since 0 inf g(K) and k> 1 + 2/c, it follows from Lemma 3 that Ill—gil c. A well known variant (and corollary) of the separation theorem substitutes compactness for interior 151: If B and C are disjoint convex subsets of a locally convex space, with B compact and C closed, then there exists a hyperplane which
strictly separates B and C, that is, there exists / in E * such that supf (C) < inf 1(B). This result can be improved if E is a Banach space, giving the following corollary of Theorem 2. COROLLARY 1. If B and C are disjoint convex subsets of a Banach space E, with B compact and C closed, there exist x in C and g in E * such that g(x) = supg(C) < infg(B). An obvious corollary of the above variant of the separation theorem states that a closed convex subset of a locally convex space E is the intersection of all the closed half-spaces which contain it, that is, if xØC then there exists
f in
and a real number c such that the half-space H {y:f(y)
c} con-
tains C but not x. We say that the half-space H supports C if C C H and
THE SUPPORT FIJNCTIONALS OF A CONVEX SET
for som€. y in C. from Corollary 1.
f(y) =
c
COROLLARY 2.
31
The proof of the following corollary is immediate
If C is a closed convex subset of a Banach space, then C is
the intersection of all the closed lzalf.spaces which support it. CORoLLARY 3. If C is a closed convex subset of a separable Banach space E, then C is the intersection of a countable number of its supporting closed half.
spaces.
Let be a dense subset of E '— C and let be the distance to C. By first applying the separation theorem to C and + then applying Theorem 2 to C and U, we can find in E* such + supports C and sup that < for n = 1,2,3, + PROOF.
from
Suppose, now, that x £ E C and let d be the distance from x to C. Choose
sup g,,(C) and completes the proof.
> x,,
II
which shows
If C is bounded, then every f in E is bounded on C, so we obtain another corollary to Theorem 2. C is a bounded closed convex subset of a Bonach space E, then the support functionals of C are dense in E. If fe E*, we say that f attains its norm provided there exists x in U such that = f(x). Since U is closed and convex, the proof of the following corollary is immediate from the previous one. (This result was first proved in (1).) COROLLARY 5. If E is a Banach space, then the set of f in E* which attain their norm are dense in E
In all the above results we could have dropped the hypothesis that E be a complete normed space provided we assumed that C itself be complete. This follows from the fact that we applied Lemma 1 only to the set C. The following 'result shows, however, that Corollary 4 (and hence Theorem 2) must fail in an incomplete space. THEOREM 3. If E is an incomplete normed linear space, then there exists a bounded, closed convex subset C of E having nonempty interior such that the support functionals of C are not dense in
E a dense subspace of its completion F and identify (in the obvious way) E and Since E * F, there exists x in F'— E such that lix Il = 1. By applying the support theorem to x and the unit ball of F, we can find I in F such that II f II = 1 = f(x). Let D = {y: ye F, ii y II 1 and f(y)=O} and let C' in Fbe the convex hull of D and x, so that C' is the set of all elements of the form z=Ax+(1—2)y, where yeD and Ae[O,1]. It is easily verified (using the compactness of [0, 1]) that the convex set C' is closed; it also has nonempty interior. (If liz — (1/2)xli (4/3) — (1/3)[1 —f(x)) — (1/3)f(x) = 1. Since x€ A and y is an interior point of A, we conclude that (3/4)y + (1/4)x z is an interior point of A. But z is also in C, so there must exist a point w in A which is in the interior of C. This is impossible, since if w€ A then g(w) ugh, while g(w) is less than at the interior points w of C. The first example given above shows that the conclusion to Theorem 2 fails if X is unbounded, even though it is assumed that C itself is bounded and X is a linear variety (i.e., a translate of a linear subspace). If we assume that X is a finite dimensional variety, however, we get a valid result (which is actually a corollary to Theorem 2). More generally, we can assume that X is a reflexive variety, that is, X = x + M for some x in E and some subspace M of E, where M is a reflexive Banach space under the induced norm. COROLLARY 6.
Suppose that C is a bounded, closed convex subset of the Banach
space E, that X is a reflexive linear variety in E, that i > 0 and that for I in hILl = 1, we have supf(C) < inf f(X). Then there exists g in E5, <e. II gil = 1, and x0 in C such that sup g(C) = g(x0) < inf g(X) and Ill PROOF.
Since f is bounded below on X, it must be constant on X; writing 0. Let E1 = ElM; under the
X = x + M as above, we then have f(M) =
THE SUPPORT FUNCTIONALS OF A CONVEX SET
usual factor-space norm, B1 is a Banach space and I can be regarded as an Let C1 be the image of C in E1 under the element of norm one in canonical mapping of E onto E1, and assume for the moment that C1 is closed in E1. Regarding X as a point in E1, we have supf(C1) 1(X), so by Theorem 2 there existsg in ugh = 1, and X, in C1 such that supg(C1)=g(Xo) 0 and assume that the result is true for subspaces of deficiency n — 1. Let g be a nontrivial functional in N5 such that g(x0) = supg(C fl N). Note that if C c N, then any functional which vanishes on N supports C at x,, so we can assume that there exists an element y in C '— N. Let N' be the linear span vanishes on each
closed convex set C such that
I
f
of Nand y, and let C' be the convex hull of CnN' and Then C' has nonempty interior relative to N', C' contains C fl N',
g(x0)).
34
ERRETT BISHOP AND R. R. PHELPS
and x0 is in the boundary of C'. By the support theorem, then, x0 is a support point of C' in N' and therefore it is a support point of C fl N' in N'. By the induction hypothesis, x. must be a support point of C. THEOREM 4. Suppose that N is a closed subspace of finite deficiency in the Banach space E, that C is a closed convex set in E, that e > 0, and that z in of C such N is in the boundary of C. Then there exists a supPort point that x,,eNond JJz—x011 <s. Paoo,. There are several cases to consider. First, if C is contained in a proper closed subspace of E, then any functional which vanishes on this subspace supports C at each of its points, so z itself is a support point of C. Assuming that C is not contained in a proper subspace, we consider whether z is in the boundary (relative to N) of C fl N. If it is, then by Theorem 1
there is a point x0 in N which is a support point of C fl N such that liz — x0 <e. By Lemma 4, x, is a support point of C in E. Finally, suppose z is not in the boundary of C fl N; then there exists a neighborhood (relative
to N) of z which is contained in C fl N. We will show that z itself is a support point of C in E. There exists a point y in E such that the segment this follows is contained in from the fact that N has finite deficiency and that z is in the boundary of C.
Let N' be the linear subspace spanned by N and y, and note that N is a hyperplane in N'. We will show that N supports C fl N' at z, and hence (by Lemma 4) z is a support point of C in E. It suffices to show that the open half-space {x + ry: xe N and r > 0) in N' is disjoint from C. Suppose that C contained a point x + ry of this half-space. Since z is in the interior (relative it such that to N) of C fl N, there would exist a point w of C and A in z Ax + (1 — 2)w. Hence the triangle with• vertices 2, W, and x + ry would a point of [y, 4, a be in C and (as can be easily shown) would contradiction.
The apparent duality between this theorem and Corollary 6 leads one to conjecture that the theorem might still be true if it is merely assumed that E/N is reflexive (rather than finite dimensional). The theorem fails, however, under this weaker hypothesis; there is a well known example (see, e.g., [2, p. 160]) of a compact convex set C in a Hubert space H and a line N in H such that C n N = {ø), but 0 is not a support point of C (even though H/N is reflexive).
All four of our lemmas have valid analogues in more general topological vector spaces, although we do not know whether this is true of the theorems themselves. (These questions will be the subject matter of another paper.) For instance, it is unknown whether a closed convex subset of a complete. locally convex space must have any support points. (For those special classes of closed convex sets C which are known to have support points, it is known that the support points are dense in the boundary of C, e.g., if C has nonempty interior, or if C is locally weakly compact [5].)
THE SUPPORT FUNCTIONALS OF A CONVEX SET BIBLIOGRAPHY
1.
E. Bishop and R. R. Phelps,
A proof :'iat every Bctnach apace is sub reflexive, Bull.
Amer. Math. Soc. 67 (1961), 97-98. 2. N. Bourbaki, Espaces vectoriels topologiques, Chapter V, Hermann, Paris, 1955. 3. N. Dunford and J. Schwartz, Linear operators, Part 1. lnterscience, New York, 1958.
4. R. C. James, Reflexivity and the supremum of linear functional,, Ann. of Math. (2) 66 (1957), 159-169.
5. V. L. Klee, Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466. 6. , Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. 7. R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976-983. UNIVERSITY OF CALIFORNIA, BERKELEY
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS BY
HARRY CORSON' AND VICTOR KLEE2 0. Introduction. This paper contributes to a problem discussed earlier by Keller [15], Stoker [271, and Klee [17; 18; 19; 21; 22]—that of classifying topo-
logically the closed convex subsets of important normed linear spaces. (A forthcoming paper by Bessaga [4] also has some points of similarity with the present work.) Our section headings are as follows: 1. A reduction theorem; § 2. Spaces homeomorphic with their positive cones; § 3. Spaces homeomorphic with their closed convex bodies; § 4. Convexity and uniform continuity; § 5. Additional remarks and problems. A convex body is a convex set which has an interior point. By the principal result of § 1, the topological classification problem for closed convex bodies in a normed linear space E is reduced to that for E's unit cell {x E: II XII i} and E's closed linear subspaces of finite deficiency. A corollary asserts that if E admits (for each finite n) a closed linear subspace of deficiency n which is homeomorphic with its own unit cell, then E is homeomorphic with all its closed convex bodies.
The support of a real-valued function! is the set suppf= {b€dmnf:fb * 0). A subset B of a normed linear space E is a substitutive basis for E provided the following three conditions are satisfied: Si—B is an unconditional basis for E;' S2—whenever x,y,zEEwith IIYII = lizit and suppf5 = suppf c /3—
then IIx+yIt=IIx+zII; S3—IIbII = 1
for all bEB.'
As Dr. Bessaga has kindly informed us, a result of Bohnenblust [5) implies
that if B is a substitutive basis for a normed linear space E of dimension then there is a linear isometry which (for some set X) carries E into the I'X or c0X and carries B onto the canonical substitutive basis for this space. However, in most instances we work directly with the substitutive property rather than using the representation afforded by Bohnenblust's theorem. § 2 shows that if B is an infinite substitutive basis for E and E is either complete or is the linear extension of B, then E is homeomorphic with its positive cone {x E: 0) by means of a transformation which is norm1
Research
supported in part by a grant from the National Science Foundation, U.S.A.
(NSF-G10738). 2
Research
supported in part by a fellowship from the Alfred P. Sloan Foundation
and in part by a grant from the National Science Foundation, U.S.A. (NSF-G18975). 2 That is, each point x C E admits a unique representation in the form z = where is a real-valued function on B and indicates unordered summation. (Of necessity supp is countable.) The normalizing condition S3 is not essential but makes for simpler notation. 37
HARRY CORSON AND VICTOR KLEE
38
preserving and positively homogeneous. It seems probable that every infinite-
dimensional normed linear space is homeomorphic with all its closed convex bodies, though this has been known previously only for reflexive spaces [21]. It is established in § 3 for several other classes of spaces, including separable conjugate spaces, Banach spaces with unconditional bases, and normed linear spaces are homeomorphic spaces. Since all 1231, this settles the topological classification problem for p0-dimensional closed convex bodies. At the same time, our method yields a new proof (independent of the rather complicated isotopy considerations of [17]) of the basic fact that Hubert space is homeomorphic with its unit cell. 1,2, and 3 fail (or their inverses fail) Many of the homeomorphisms of to be uniformly continuous. The results of § 4 show that this is not accidental; in particular, a bounded convex subset of a normed linear space does not admit a uniformly continuous mapping onto any unbounded metric space. § 5 discusses some unsolved problems and indicates still another way of proving that Hubert space is homeomorphic with its unit cell. 1. A reduction theorem. The homeomorphisms which we seek are obtained from the composition of several simple transformations. Recall that for a point p of a convex set C, the characteristic cone of C relative to p is the set cc (C, p) = (x: p + [0, co[(x — p) c C}. This is the union of all rays from p which lie in C, except that cc (C, p) = when there are no such rays.
1.1 PROPOSITION.
are (for c =
Suppose
and E6 are topological linear spaces, W. and Z,
E1, and pt is a point of E, such that p,€int W1c W1cintZ, and cc(W1,p1)=cc(Z,,fr1). Then every homeomora, b) closed convex bodies in
mt phism of OZ, onto OZb can be eltended to a homeomorphism ij of Z. and zeJp.,z]—int W.. such that onto Z6— mt
(Here 0Z1 is the boundary of Z1.) PROOF. For each z E 0Z1, the segment [ps, z] intersects the boundary 0W1 in a unique point w,(z). The assumption about characteristic cones implies that the set Z. — mt W1 is simply covered by the set of segments z let map the segment [wa(z), zJ affinely onto the segment z
carrying the endpoints in the indicated order. Then c and mt WQ biuniquely onto Zb — mt W6. That 7) is a homeomorphism can be established by a straightforward geometric argument. Alternatively, let P1 and v, denote respectively the gauge functionals of W1 and Z1 relative to p4. Then each point x E Za mt W0 admits a unique expression in the [wb(Ez),
carries Za
form x = (1
+ Ag(x/vax) with
—
]0,
11
and we have = Since
(1 —
pb(e(xlMax))
+
all the gauge functionals are continuous and since
-
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
= (v,,xXp,x —
39
—
it follows that is continuous. Similarly, if' is continuous and the proof of (1.1) is complete. 1.2 PROPOSITION. Suppose M is a closed linear subspace of infinite deficiency in a nornzed linear space E and J is a closed halfsfiace in E whose bounding hyflerJ.ilane H contains M. Then there is a homeomorphism of J .— M onto J which carries H — M onto H.
Let u denote the natural homeomorphism of F! onto its quotient Since HIM is an infinite-dimensional normed linear space, a theorem in [19) guarantees the existence in HIM of a sequence C, of unPROOF.
space HIM.
bounded but linearly bounded closed convex bodies such that 0€ mt C,, always mt C,, C,,+1, and nrc,, = 0. For each n let p,, be a point which is interior to the set tr'C, relative to H. Let z E J — H and for each n let B,, u'C,, + [—1/(n + 1), l/(n + 1)]z. Then the following assertions are easily verified: for each n, fi,, is interior to the closed convex set B,,; always mt B,,
cc(B1,p,,) = /',, + M for all i
flrB,.= 0.
Let e be a positive number such that B0' contains the 2eneighborhood of M and for n 1 let denote the closed em-neighborhood of M. Let = 0 for all n. Then the conditions displayed above for B,, and fi,, are satisfied also by and = M rather than 0. except that Let h0 be the identity mapping on the set E mt B0. Then h0(0B.) = OBO', and from 1.1 it follows that the homeomorphism h0 I can be extended to a homeomorphism h, of B0 — mt B, onto BJ mt which takes points of into H, J—H into E—J. (Recall that Now let B0' = B0.
Continuing in this way, we obtain a sequence of homeomorphisms h0, h,, h2,-.such that h,, carries B,,_, —. mt B,, onto mt h,, extends h,,-, and each of the sets J H, H, and of E J is carried into itself by the transformation h = Since nr B,, = 0 and nr B = M, it follows that h is a homeomorphism of E onto E -.' M which carries J onto J — M and H onto H M. Then h' If — M is the homeomorphism sought for 1.2. We now state 1.3 THEOREM. Suppose C is a closed convex body in a normed linear space E, 0€ mt C, and L = (XE E: (0, oo[x c C}, a closed convex cone with vertex 0.
Then if L is a linear subspace of finite deficiency n in E, there is a ho,neomorphism of C onto the product of L by an n-cell which carries the boundary 0C onto L x If L is not a linear subspace or is a linear subspace of infinite deficiency, there is a homeomorphism of C onto a closed halfspace J in E which carries OC onto the bounding hyperplane 8J; when E is infinite-dimensional, there is also a homeomorphism of C onto the unit cell U = (x E: x S 1) which carries i3C onto the unit sphere 8 U tx E: II x II = 1 }. PROOF. In view of the results of [17J it suffices to consider the case in
40
HARRY CORSON AND VICTOR KLEE
which L is a linear subspace of infinite deficiency in E. Suppose first that L = (0), whence C is linearly bounded, and for each x E 8C let Ex = x/I I x I I e 8U. Then E is a homeomorphism of OC onto OU, and by using 1.1 it is easy to extend e to a homeomorphism of C onto U. Now suppose dim L 1. Let H be a hyperplane thiough 0 which does not contain L and let J be one of the closed haifspaces bounded by H. We shall onto produce a homeomorphism C of C onto f— (L Ii H) which carries He— (L fl H). This will suffice to prove the Theorem 1.3, for by 1.2 and a part of the Theorem already proved [17] we are assured of the existence of (L fl H)) = (L fl H)) = homeomorphisms f and g such that H,gJ= U, and gH=0U. Let z€L—H, so that each point x€E admits a with x1 H and x2 R (the real unique expression in the form x = x, + number space). Let V = {xe E: 1k1 II fl is the given norm in E. For each xeE, let C1x = 11(1 + IIxIDx. Then by a result on p.
is a homeomorphism of E into E and C1K is convex whenever is a bounded convex homeomorph be of It is clear that 0 E mt C,C and (ci C,C) C1C = (0C1C) L. Let the homeomorphism of cI C1C Onto V which, for each ray p from 0, carries the segment p fl cl C1C linearly onto the segment p fl V. Then C?AC = V (L fl 8 V) and CZIOC =8 V L. Let denote the "stereographic projection" of V (z} onto the set J' = H + [0, lIz. Specifically, for each ray p which emanates from z and intersects H, carries the segment p fl V affinely onto the segment p ti I'. Then is a homeomorphism of V (z} onto J' which carries the set C2C1C onto J' (L fl H) and carries W.IOC onto H — (L fl H). For each x = x1 + x2z€J', let C4x = x1 + Then the transhas all the desired properties, and this completes the formation C = proof of Theorem 1.3. Note that two of the three possibilities for L in 1.3 were treated completely on pp. 30-31 of [17]. However, previous treatment of the case in which L is a linear subspace of infinite deficiency depended on the existence of a continuous linear projection of E onto L (17] or at Least the existence of a continuous inverse for the natural homomorphism of E onto E/L (21). By a theorem of Bartle and Graves 12), such an inverse exists when E is complete, but an example in 120] shows that it may fail in general. 42:of [18],
K is convex and 0€ K. Thus the set
1.4 CoRoLLARy.
If C is a closed convex body in an infinite-dimensional
normed linear space E and C contains no line, then C x [0, 11 is homeo,norphic with C x (0, 1].
PROOF. We assume without loss of generality that 0 e mt C.
The space
E x R is normable, C x [0, 1[ is homeomorphic with C x [—1, oo[ and C x [0, 1) with C x [—1, 1]. The sets C x [—1, co[ and C x [—1, 1) are closed convex bodies in E x R which have (0,0) as an interior point. The characteristic cone of C x [—1, oo[ is not a linear subspace and (since C contains no line) that of C x [—1, 1] is either not a linear subspace or is equal to ((0,0)), a subspace of infinite deficiency. Thus the desired conclusion follows from 1.3. The following remark is easily verified:
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
If F1 and F2 are closed linear subspaces of the same finite deficiency in a topological linear sPace E, there exists a linear of E onto E which carries F1 onto F2. If E is a normed space there is a positively homogeneous norm-Preserving homeomorphism of E onto E which carries F, onto F2. We conclude the section with 1.6 COROLLARY. Suppose the norined linear space E admits (for each finite n) a closed linear subspace of deficiency n which is homeomorphic with its own unit cell. Then E is homeo,norphic with all its closed convex bodies. Pnoor. In view of 1.3, it suffices to establish the following for each n: Se—If F is a closed linear subspace of deficiency n in E, then E F x [0, 1]" means "is homeomorphic with"). Obviously S0 is true. Now suppose Sn-, is known and consider a closed linear subspace F of deficiency n. Choose x€E—F and let G = F+ Rx, a closed linear subspace of deficiency n—i. 1.5 PR0I'oslrIoN.
Let C and D be the unit cells of F and G respectively, whence (using 1.5 F and D G. It follows that
and the principal hypothesis of 1.6) C
[0, 1]
[0, 1)
whence
and
(using the inductive hypothesis) (F x [0, 1]) E G x [0,
F x [0,
x [0,
the proof is complete.
2.
Spaces homeomorphie with their positive cones. When B is an uncondi-
tional basis for a normed linear space E, each point x E admits a unique It will often be convenient to supexpression in the form x = press the distinction between x and f1, writing supp x for for fib, The following stronger "substitutive" property of substitutive bases follows easily from their definition and trivially from the representation theo• rem stated in the Introduction. 2.1 PROPOSITION. Suppose B is a substitutive basis for a normed linear space E, and w,x,y, and z are points of E. If IlwII IIxILIIylI Itzll, and
etc.
(suppw
U suppx) fl (suppy U suppz)
=
0,
then In order to describe an auxiliary transformation which plays an important
role in the proof of 2.3, we define a fan to be a closed convex set F c such that F is symmetric with respect to the line {(r, s): r = s}, and (1, 0), (0, 1')} c Fc cony 0), (1,0), (0, 1), (1, 1)} cony The gauge of F is
the function i-,
on R2 which
s) = inf lm > 0:
Thus tions
(I r
is
defined as follows:
j, IsI) mF}
is a norm for Rt whose unit cell is the union of F with F's reflecthrough the origin and across the two axes. We say that the fan F
HARRY CORSON AND VICTOR KLEE
42
can be spread provided there exists a mapping ço of the quadrant Q = such that the following ((r, s): r 0 s} onto the halfplane {(r, s): $ 01 conditions are satisfied:
ç(1,O)=(1,0)
10
and 9'(O,l)=(—l,O);
ço is positively homogeneous; 3° is norm-preserving (i.e., on Q); = 4° is distance-increasing (i.e., r,(q'x — coy) r,(x — y) for all x, y E Q); — coy) is Lipschitzian (i.e., there exists m < oo such that 5° 2°
— y) for all x, y Q). Each mapping which satisfies these five conditions will be called a spreading of F. çø
We have been unable to determine whether every fan can be spread. However,
the following can be proved:
2.2 PROPoSITIoN. Let F be a fan and P = {(r, s): r,(r, s) = 1). SupPose there exist a subset Zof 1', an arc A in Z and a number m< such that F—Z
is finite, 1' has finite curvature kz at each point z C 2, and sup kA m inf kZ. Then F can be spread. The proof of 2.2 will be omitted, as it is rather tedious. However, some important special cases are easily treated. For example, suppose F is the fan v, is the Euclidean norm for R2. Let lJ, so that + 1(u, v): u 0 ço denote the mapping which sends the point v = (r, 0) (polar coordinates) in the first quadrant into the point v' = (r. 20) in the upper halfpiane. Clearly .p satisfies conditions
1° —
r,(P
for
—
3°, and we claim that q)2
—
q'? 4r,(P —
p and
q in the first quadrant. Because of the homogeneous and rotational nature of the transformation, it suffices to consider the case in all points
which q = (1, 0), and then for p
I + r2— to be established
= (r, 0) the above inequalities reduce to
1+
+4r2—8rcos0,
for all r 0 and 0€ [0, But then the first inequality is equivalent to the assertion that 3(1 — r)2 + 4(1 — cos
obvi-
ous and the second is
(This argument, due to Branko GrUnbaum, is simpler than our original
0.
Two other cases of special interest are those of the fans F for which
r,1(u,v)=IuI+lvI and rr,(u,v)=max(luI,IvD. These fans can be spread as
for 0
follows: For p = (r, 0) in let for
n/2.
the first
quadrant,
and for
let çc'(p) = (r, 20);
(Here r represents the r,,distance from 0 and 0 is
the usual
Euclidean angle.)
From 2.2 we see that the following result applies to the spaces for each infinite cardinal Alternatively, the proof for reduced to
that for
and
can be
by means of Mazur's homeomorphism (241.
2.3 Suppose B is an infinite substitutive basis for a normed linear space E, and E is complete or is the linear extension of B. Then there exists
43
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
which carries the
a positively homogeneous norm-preserving homeomorphism 0) onto the entire space E. positive cone {x E:
Let PA = {x e E: 0 for all a A). Let us assume that E is complete and denote by L the linear extension of B. We shall define a positively homogeneous norm-preserving
For A c B and xe E, let x4 = LEA
PROOF.
(as well as distance-increasing and Lipschitzian) homeomorphism of L fl PB onto L and then show that it can be extended to carry PB onto E. Let 'p and for x€P{a,b), define be a spreading of F. For a,beB with a +
= where So(xa, Xb) =
P(b}
xb)a + 502(Xa, x6)b
Xb), 502(xg, Xo)) £ R2.
,
is positively homogeneous,
Since
Since 'p is norm-preserving and B is a substithe same must be true of is distancetutive basis, is norm-preserving. Similarly, it is clear that increasing and Lipschitzian. The desired homeomorphism 1) wiLl be described in terms of the transformations We shall discuss only the case in which B is countable since the general case can be handled by a simple extension of this reasoning. be an enumeraLet I denote the set of all positive integers and let tion of the members of B. For each i, let denote the transformation of onto For XE PB, let e1x and then proceed = inductively to define x for I e I (1). We shall write x, for
x L fl PB and k is the largest index for which I k. Thus for x 6 L fl PB, we may define (Although the are defined on all of PB, it is not obvious
x
* 0, then
=
7)kX for all
= that this limit exists when XE PB L.) Since is a positively homogeneous norm-preserving honieomorphism of onto P1o,4-d, it is easy to verify that 'j is a positively homogeneous norm-preserving biunique transformation
of L fl PB onto L. We wish to extend i, to the entire positive cone PB (which is the closure of L fl PB) and then to prove that the extended is a homeomorphism of PB onto E. The following fact is crucial: 10. Suppose XE PB and Then the sequence x,ô. for each n €1. is
a Cauchy sequence.
To prove 10 we consider Xe PB, m €1, and n €1 with m > n, and check that the following equalities and inequalities are valid where A is the Lipschitz constant for so and hence for all II
— 'ix" II = II 'i,.x"' —
I
I
:z
I )?nX" — 'id'
+ —
+
(1),1X — 7JnX )nôn + (m,x
+
—
= II
i-'ni.3
+ 1k" —
+
—
" — _4A J,II Ijmn—IX —
fl\2
W(\ I + k'i,,—,X ),'+i°n+I
*tI II —$AIIX,,+IUa+IIVT,X—X —
I
I
II
+ lix" —
I + X — X11+1 f
I
II
HARRY CORSON AND VICTOR KLEE
whence clearly lim,,,-..., — 'ix" I = 0 and ('iX'),E, must be a Cauchy II Three of the above steps are immediate from the relevant definisequence. I
tions, and the other five can be justified as follows: = 'i,,x", for x? = 0 for i> n. Then use the fact Note first that o o •-• o and each of the transformations is normthat ij, = i,,, o preserving. affects = $ This follows from the facts that certain coordinates are zero, only the ith and (i + 1)th coordinates, and the 'i,'s are produced by composition
of the
Note that x? = x? for 1
i n, and that the transformations
affect
only certain coordinates. and relevant definitions. Refer to 2.1, the Lipschitzian nature of and 1' are alike in all coordinates affected by ij,,—,. Note that Next we claim flu. For each x PB, — I = 0. in fact, (I — 'i,x"
I
=
+
—
=
—
+
+
II ('i,,x —
+
—
II
All
+
—
II
+ lix — x"" II
+ lix — x"'t Ii which clearly converges to 0 as n —.
'ix" e E. We shall establish Now for each x e PB, let 'ix = is a Cauchy sequence in L fl PB, then 'i(Xi),E, is a Cauchy 111°. if sequence in L. Let x = urn1-.., xi PB. Then for each i e / and n e i, ii'ix — vixill
— 'ixiiI + li'ix —
and
U'ix" — 'ixill = lim,....ii'i1x" — il'i1,x — 'i,,xiJl +
'if
'ix — to), — whence 111° follows.
= li'i,,x" — 'i,,xili —
= 0 by the definition of 'ix (which was justified by A"(Ix — xiii, = 0 by 11°, and of course ii'i,,x
From 111° we conclude that 'i (defined now on all of PB) is a continuous transformation of PB into E, and of course 'i(L fl PB) = L. Clearly '7• is norm-preserving and positively homogeneous. Since each transformation increases distances, the same must be true of 'i and consequently 'i is biunique is continuous. Thus to complete the proof of 2.3 it remains only to and show that i'jPB E. This may be done directly or by observing that 'iPB, being homeomorphic with the complete space PB, must be a Ga set in E 126], and that a linear subspace which is a C3 set must be closed [25]. This dis-
poses of 2.3 for the case in which B is countable.
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
45
Spaces homeomorphic with their closed convex bodies. 3.1 THEoREM. Each of the following conditions on a normed linear space £ insures that E is homeomorphic with all its closed convex bodies; 3.
£ is E admits an infinite substitutive basis B whose linear extension is E; E is a Banach space and some closed linear subspace of E admits a continuous linear transformation onto a Banach space homeomorphic with the space I. PRoOF. In view of 1.6, it suffices to prove that whenever F is a closed
linear subspace of finite deficiency in E, then F is homeomorphic with its own unit cell. With the aid of 1.5, we see the sufficiency of proving that each space £ (as described above) is homeomorphic with its unit cell. In the third case, this follows from the theorem of [21]. For the second case, choose b e B and let H denote the hyperplane {x E: Xb = OJ. Since B — {b} is a substitutive basis for H, it follows from 2.3 that and Since clearly PB—(HflPB)x [O,oo[ it follows that E H x [0, oo[ and then from 1.3 that E is homeomorphic with its own unit cell. (Notice that this argument also applies to the completion of such an E.)
Consider, finally, the case in which E is Let C denote the set of all finitely supported members of the Hilbert space and let U and W be the unit cells of £ and G respectively. Let x. be a basis for E and the natural basis for G. From the second case it follows that C— W.
We know from [23] that E G and wish to show that U
W. Since E is
separable, it can be strictly convexified [6]; that is, there exists in E a symmetric closed strictly convex body U' whose gauge functional is a norm which generates the given topology in E. Of course U U' by means of a simple "radial" homeomorphism. Now Xr, and z0 are B-systems (in the sense of [12; 23]) for E and G respectively, as normed by the gauge functionals of U' and W, so the associated Kadec-Bernstein mapping [12; 231 is a homeomorphism of E onto C which carries U' onto W. Reviewing the assembled facts, we deduce that U W and complete the proof of 3.1. 3.2 COROLLARY. Each of the following conditions on an infinite-dimensional Banach sPace implies that E is homeomorphic with all its closed convex bodies: E is a closed linear subspace of a Banach space which admits an unconditional
basis;
E is reflexive;
E is a
norm-separable subspace of a conjugate space;
E is a closed subspace of finite deficiency in a space CX for some infinite compact Hausdorff space X. PROOF. For the first two cases, let S be an infinite-dimensional separable closed linear subspace of E. Then S is homeomorphic with I by results of Bessaga and Pelczyñski [3] and Kadec [13], so the desired conclusion follows
from 3.1. In the third case, E itself is homeomorphic with I by a theorem of Kadec [14] and Klee [20]. In the fourth case, we may assume (in view of
HARRY CORSON AND VICTOR KLEE
46
cX. Then = = O}, where 1.5) that E= {çoeCX: çpx1 çox, = (adopting a suggestion of M. Jerison) since X is infinite there exists in X an If cY=X infinite sequence of distinct points such that ci is a discrete subset of V. i 2} Otherwise, we choose a then of y10 in the complement of some neighborhood of and subsequence treat in a similar manner. Continuing in this way, we find that either j 1} is discrete. In either i 2) is discrete for some j or the set case, there exists an infinite sequence z0 of distinct elements of X such that is discrete. For each E CX, let vjço = Z. Then of the set Z = U course is a continuous linear transformation of CX into CZ, and it is easily includes every function on Z such that —, 0 and Cx. = 0 verified that for all i. Thus liE has a subspace equivalent to c0, and since c0 is homeomorphic with I by a theorem of Kadec [12], the desired conclusion again folI
lows from 3.1. 4. Convexity and uniform continuity. The basic idea of this section is apparently due to Efremoviè [81 and was suggested to us by John Isbell.
Some related ideas are employed by Edeistein [7). An t-chain in a metric space (M, p) is a finite sequence (Po, of points pt) < e. Such a chain is said to join Po and of M such that always A subset S of M will be p,1, and its length is the number , p.). called quasiconvex provided for each ö > 0 there exists 4'a < oo such that any
two points x and y of S for which p(x, y)> ö can be joined (for each by an i-chain in S of length at most çbip(x, y).
0)
4.1 PRoPosITIoN. Each of the following conditions on a metric set S implies that S is quasiconvex: a°. S is a complete metric space which is metrically convex; b°. S is a convex subset of a normed linear space; c°. S is the boundary of a bounded convex body in a normed linear space; d°. S is a bounded starshaped subset of a normed linear space. PROoF. In cases a° and b°, each two points x and y of S lie together in a subset of S which is an isometric image of the interval [0, p(x, y)]. Thus S is quasiconvex with cl's = 1 for all ö> 0. For case c° we shall employ the following fact from [16]: (t) Suppose Y and Z are bounded closed convex bodies in a normed linear space E, and 0€ mt (Y fl Z). Let p denote the norm of E, rj the gauge functional of Y, C the gauge functional of Z, and (for all x€ E) li*x = max (lix, li(—x)),
Let r denote the radial mapping of E onto E which carries Y onto Z, so that tO = 0 and rx = (iix/Cx)x (or x E 10). Then for all u, tie E, p(vu — rv) Mp(u — v), where Cix = max (Cx, C(—x)).
M=
+ C
p
+
C/\
P
Now suppose C is a bounded convex body in a normed linear space E, and assume without loss of generality that 0€ mt C. With U denoting -the unit cell
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
47
freE: lixil 1}, there are numbers >0 and < oo such that A1Uc Cc A,L1. Consider two points x and y of the boundary OC. We shall prove that for each e > 0, x and y can be joined in OC by an e-chain of length no greater This is enough for our purpose, though doubtless the than (2 + result can be improved. Let H be a two-dimensional linear subspace of E such that {0, x, y} c IT. By a simple and well-known argument, the plane IT must contain an elliptical disc D centered at 0 such that D c U fl IT c i/iD. Then of course A1D c C c i/TD. Let r denote the radial transformation of IT onto II which carries C onto D. By the result (t) we see that for all u, v e IT it is true that IIru — rvil klIu — vu and
hr the norm in E, k — (2 + VT)/A1,
where liii is and k' = 2 V'TA2 + V2A/A1 Now consider an arbitrary e > 0 and points x and y of OC. Let p denote the
of the elliptical disc D, a norm for if. Clearly rx and ry can be joined in the ellipse OD by an e/k'-chain Po, whose p-length is at most (ir/2)p(rx — ry). But then the points form , an e-chain joining x and y in 0C, and the II Il-length of this chain is gauge
k'
—
Pi)
rIjTX—ryll This completes the discussion of case c°.
For case d°, let P be a point from which S is starshaped and let r = sup,€3 II s —
P11
x,yeS with
< oo. Consider an arbitrary ö > 0.
of length lix — length
I
I P — y II
It is clear that whenever then x and p can be joined in [x,p] by an echain r and p and y can be joined in [p,y] by an e-chain of
P11
r,
—
An e-chain (p0, chain.
y
whence x and y can be joined in S by an e-chain of length II. •,
The proof of 4.1 is complete. consisting of n + 1 points will be called an (n, e)-
4.2 PsoI'osliION. A metric space (M, p) is quasiconvex if and only for each .3> 0 there exists $, < oo such that any two points x and y of S for which p(x, y) >.3 can be joined, for each e e JO, 1(, by an (n, t)-chain in S with ne 198p(x, y) PROOF. Since the length of an (n, is at most ne, the aiffl part is obvious. Now suppose (M, p) is quasiconvex, .3 > 0, x and y are points of M with p(x, y) >.3, and (q0, - - ., q,,) is an e/2-chain in M which joins x and y and is of length (y Let r0 = 0; and having chosen r0 < r1 < let be largest integer k n such that Since , qj < e. (q0, . . •, > e/2 is an e/2-chain, it is clear that r,+1 > r and that
HARRY CORSON AND VICTOR KLEE
48
< n. Lets be such that r= n andletpi=QYL (/'0, , p,) is an (s, e)-chain joining x to y, and since j s — 1, we have when
(s — 1)
j-
,
p)
p(q,_1 ,
Then 1 > e/2 for 1
#ap(x, y),
2çt'ap(x, y) + e (2çbc + 1/ô)p(x, y). This completes the proof of 4.2. A mapping jp of a metric space (M, p) into a metric space (M', p') wilt be called Lipschitzian for large distances [resp. Lipschitzian for small distances) prosuch that p'(çox, coy) Asp(x, y) whenvided for each J> 0 there exists 4 ever x, ye M and p(x, y) > o [resp. p(x, y) < 8]. Of course, 'p is Lipschitzian if and only if 'p is Lipschitzian for both large and small distances. It is evident
whence Sc
Lipschitzian for small distances, then 'p is uniformly continuous. Suppose (M, p) is a quasiconvex metric space and 'p is a 4.3 (M, p) into a metric space (M', p'). uniformly continuous transformation of Then ço is Lipschitzian for large distances and hence carries bounded subsets of
that if 'p
is
M into bounded subsets of M'. PRooF. The uniform continuity of so is not used in its full strength. Indeed, it suffices to assume the existence of a number £ JO, and a number < b whenever p(u, v) e. Consider an arbitrary b < oo such that p'(çou, o >0 and let be as in 4.2. Then if x andy are points of M with p(x,y) > O,x
can be joined to y by an (n, e)-chain (p0, But then p'(çox,'py)
.,
nb
in M with ne
y).
PA p(x,y)
whence ço is Lipschitzian for large distances.
Now consider a subset X of M whose p-diameter is a number d < co an arbitrary pair x and y of points of X. Then when p(x, y) e
Consider
we have p'('px,
soy)
e, p'(çox, coy)
p(x, y)
Consequently the p'-diameter of çoX is at most b max (1, 194/e), and the proof
of 4.3 is complete. 4.4 COROLLARY. ij X is a bounded convex subset of a normed linear space, or is the boundary of such a set, then every uniformly continuous metric image of X is bounded.
(Still open, apparently, is Gorin's question LiOl as to whether Hubert space is uniformly equivalent with some bounded subset of itself.) 4.5 COROLLARY. Suppose Y is a convex cone with vertex 0 in a normed linear space, and ço is a positively homogeneous transformation of Y into a Added in p,oof: Professor Isbell has pointed out the similarity between this result and a theorem of Atsuji 101.
TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
normed linear space. If ip
is
49
uniformly continuous, then ço is Li/.'schitzian.
5. Additonal remarks and problems. Although we conjecture that every infinite-dimensional normed linear space E is homeomorphic with its unit cell C, we cannot prove even that E x [0, ii E x [0, IL. (By 1.4, C X [0, 1)
C x [0, 1[.)
It would be especially interesting to extend Theorem 2.3 to the case of a Schauder basis, though this would probably require new techniques. Bessaga
and Pelczyñski [81 have proved that a separable Banach space is homeomorphic with I if it has a closed linear subspace homeomorphic with 1. They
have pointed out (in a letter) that by a theorem of James [Ii], every nonreflexive separable Banach space E admits a closed linear subspace F with a
Schauder basis such that the positive cone Q generated by this basis is homeomorphic with the positive cone P in 1. Now P 1 by our Theorem 2.3, and thus if 2.3 could be extended to show Q F we could conclude that F I, whence E 1 by [31. In conjunction with Kadec's result [13] for reflexive spaces, this would show that all separable infinite-dimensional Banach spaces are homeomorphic, thus solving the famous .problem proposed by Fréchet [9] and Banach [1]. We repeat the question: If is a Schauder basis for a Banach space E, must E be homeomorphic with the positive cone consisting of all points E for which always 0? Note that by combining 2, part of 3, and the elementary reasoning of [19], we obtain a new proof, independent of the rather complicated isotopy considerations of [171, that Hubert space is homeomorphic with its unit cell We now indicate still another proof of this fact which replaces the reasoning of § 3 and [19] by another argument similar to that in § 2. Let P denote
the positive cone and U the unit cell of the Hilbert space P and let E be the set of all finitely supported members of 12. It follows from 2.3 that P 12, P ii U U, E (1 P E, and E fl P fl U E ii U. To show that P U it suffices to produce a homeomorphism h of E fl P (1 U onto E fl P such that both h and h1 preserve Cauchy sequences, for then h can be extended to a homeomorphism of P fl U onto P and the desired conclusion follows. Since the detailed construction of such a homeomorphism is rather technical, we shall merely describe a simple "unfolding" homeomorphism which carries E fl P fl U onto E (1 P; the same idea can be employed with more care to control the Cauchy sequences as well. A typical stage of the unfolding process is described in the following 5.1 LEMMA. Suppose A, B, and C are distinct spherical cells in a Euclidean space S. concentric at 0 with A c B c C. Suppose H is a hyperplane through 0, p' is a ray from 0 orthogonal to H, and J is the closed halfspace H + p' Suppose K is a pointed closed convex cone in H with vertex 0, and K' = K + p'. Then there is a homeo,norphism of K' fl B onto K' fl C such that Ex = x whenever xe (K' fl A) U (H fl B), and E(Q fl B) = Q fl C for each "quadrant" Q of the form p -f- p' where p is a ray in K emanating from 0. -
Paoop.
Make the appropriate construction for one of the quadrants p + p'
HARRY CORSON AND VICTOR KLEE
50
and then extend it to the others by rotation of the space S about the ray p'. 5.2 PROPOSITION. If X is the space consisting of all finitely supported of 12 and P and U are the positive cone and unit cell of I, then Pfl P. PROOF. It suffices to produce a homeomorphism of P fl 2 U onto P. For n = 1,2, ..', let X,. = fx e X: x' = 0 for i> n} and let p. be the ray {x e X: x" 0, 0 for i * n}. For n 2,3, -••, let be the homeomorphism of X,. A Pa nU onto X,, n Pfl (n + 1)U which is obtained from 6.1 by the following assignment of roles for the principal actors in 6.1: S = X, A =
0), K =
Let g. be the extension of agreeing that if (x', x2, . , f, 0,0, -) dmnf., then {x
X,,:
g,,x
— —
1/
X._1
1
,
.2
,
ii P.
-., X* ,
\ u, • * ', -i- IA A
1= obtained by
.. , X11+1 , X
Finally, let and let hx=lim...,.h,,xforeachxePfl2U. It can be verified that /t is a homeomorphism of P fl 2U onto P.
0. M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958). 11—16. 1. S. Banach, Theorie dee operations lineaires, Monogr. Mat., Vol. 1, Warsaw, 1932. 2. R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413. 3. C. Bessaga and A. Pelczyfiski, Some remarks on homeomorphisms of Banach spaces, Bufi. Acad. Polon. Sci. Sbr. Sci. Math. Astronom. Phys. 8 (1960), 757-761.
4. C. Bessaga, On topological classification of linear metric spaces, Fund. Math. (to appear).
5. F. Bohnenblust, An axiomatec characterization of L,-spaces, Duke. Math. J. 6 (1940), 627-640.
6. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.
7. Michael Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10. S. V. A. Nonequimorphism of Euclidean and spaces, Lispehi Mat. Nauk 4 (1949), no. 3 (30), 178-179. (Russian) 9. M. Fréchet, Lee sep50.. abatraits, PariB, 1928. 10. E. A. Gorin, On uniforsn-topoiogiov,,Z embedding of metric space. in EucUdean and Hiibert spaces, Uspehi Mat. Nauk 14 (1959), no. 5 (89), 129-134. (Russian) 11. Robert C. James, Reflexivity and the aupremuin of linear functional., Ann. of Math. (2) 66 159-169. 12. M. I. Kadec, On homeoinorphi.sms of certain Banach spaces, Doki. Akad. Nauk SSSR 92 (1953), 465-468. (Russian) 13. , On strong and weak convergence, Dokl. Akad. Nauk SSSR 122 (1958), 13-16. (Russian)
14. , On the connection between weak and strong convergence, Dopovidi Akad. Nauk RSR 1959, 949-952. (Ukrainian) 15. 0. H. Keller, Die Homoiomorphie der keavexen Mengen im Hilbertac/ten
____, TOPOLOGICAL CLASSIFICATION OF CONVEX SETS
Raum, Math. Ann. 105 (1931), 748-758. 16. Victor Klee, On a theorem of Bête
51
Amer. Math. Monthly 9 (1953),
618-619.
17. Convex bodies and periodic ho,neoinorphisma in HiThert space, Trans. Amer. Math. Soc. 74 (1953), 10-43. 18. -, Some topological properties of convex sets, Trans. Amer. Math. Soc.
78
(1955), 30-45.
19. , A note on topological properties of normal linear spaces, Proc. Amer. Math. Soc. 7 (1956), 735-737. 20. , Mappings into normal linear spaces, Fund. Math. 49 (1960), 25-34. 21. , Topological equieolsnce of a Banach space with its unit cell, Bull. Amer. Math. Soc. 67 (1961). 286-290. linear spaces (to appear). 22. , Topological structure of 23. Victor Klee and R. C. Long, On a method of mapping due to Kadec and Bernstein, Arch. Math. 8 (1957), 280-285.
24. S. Mazur, Une remarque ear l'homeomorphie des champs fonctionnels, Studia Math. 1 (1929), 83-85.
25. S. Mazur and L. Sternbach, (Jber die Boretschen Ty pen von Zinearen Mengen, Studia Math. 4 (1933), 48-53. 26. W. Sur lee ensembles corn plots dun espace (D), Fund. Math. 10 (1928), 203-205.
27. J. J. Stoker, Unbounded convex point sets, Amer. J. Math. 62 (1940), 165-179.
AN UPPER BOUND FOR THE NUMBER OF EQUAL NONOVERLAPPING SPHERES THAT CAN TOUCH ANOTHER OF THE SAME SIZE BY
H. S. M. COXETER
Introduction. The problem indicated in the title is regarded as the case zr/6 of the n-dimensional analog of the problem of packing small circles of angular radius 4, on a sphere in ordinary space (Coxeter 1962). Reasons are given for believing that the number of such (n — 2)-spheres that can be packed on an (n — 1)-sphere has the upper bound 1.
4,
2F%_l(a)/F$(a)
where a is given by sec 2a =
sec 24, + n — 2
and the function F is defined
recursively by ir
= F1(a) = 1. This upper bound is attained with the initial conditions when the (n — 2)-spheres are inscribed in the cells of a regular polytope
in which case a = 2. Kepler, Gregory, and Hales. Kep!er (1611) discovered the cubic closepacking of equal spheres in Euclidean 3-space. Their centers, forming the face-centered cubic lattice, may be taken to have integral Cartesian coordinates
with an even sum; then the common radius of the spheres is i/i/2 Each sphere touches 12 others; for instance, the one with center (0,0,0) touches those with centers (0, ±1, ±1), (±1,0, ±1), (±1, ±1,0). These 12 centers are the vertices of a quasi-regular polyhedron, the cuboctahedron, which was described by Plato. The existence of the packing is closely associated with the fact that the circumradius of this polyhedron is equal to its edge-length. Among the unpublished papers of David Gregory, H. W. Turnbull found notes of a conversation with Newton in 1694 about the distribution of stars of various magnitudes. The question arose: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? Gregory said "Yes", and Newton said "No"; but 180 years were to elapse before r conclusive answer was given. Stephen Hales (1727) made a random close-packing of dried peas, filled the interstices with water, and kept the vessel closed. In his own words, i compressed several fresh parcels of Pease in the same Pot, with a force equal to 1600, 800, and 400 pounds, in which Experiments, tho' the Pease dilated, yet they did 53
H. S. M. COXETER not raise the lever, because what they increased in bulk was, by the great incumbent
weight, pressed into the interstices of the Pease, which they adequately filled up, being thereby formed into pretty regular Dodecahedrons.
Hales presumably reached his conclusion by observing some pentagonal faces on his dilated peas. Modem versions of his experiment have confirmed the prevalence of pentagonal faces. Bit the solid cells could not all be regular dodecahedra. For, since the dihedral angle of this Platonic solid is less than 120°, three specimens with a common edge will leave an angular gap (about 10° 19'). In other words, twelve unit spheres with their centers at the vertices of a regular icosahedron of circumradius 2 (and edge 2 sec 18° = 2.102924- -) will all touch one central unit sphere, but will not touch one another. Gregory may have imagined that, by letting the twelve spheres roll on the central one until the gaps are all concentrated in one direction, there might somehow be enough space for one more, so that the central sphere would touch thirteen others. It may reasonably be argued that there is enough space (actually enough for 13.397 spheres, as we shall see in § 9). The problem is to arrange -
the twelve spheres so as to concentrate the free space and make room for the thirteenth. The impossibility of such a redistribution was eventually proved by R. Hoppe (see Bender 1874), thus settling the Gregory-Newton controversy in favor of Newton. Simpler proofs were given by Gunter (1875), SchUtte and van der Waerden (1953), and Leech (1956). 3. Dirichlet and Barlow. To be precise, an arrangement of equal spheres
is called a Packing if no point of space is inside more than one sphere. In any packing, we may associate with each sphere a Dirichiet region (or Voronoi Polyhedron) consisting of all the points that are as near to the center of that sphere as to the center of any other. Such regions, each surrounding a sphere, fit together to fill the whole space without interstices (Dirichlet 1850; Voronoi 1908). The density of the packing may be defined as the average of the ratio of the volume of a sphere to the volume of the Dirichiet region that surrounds it. For instance, the density of Kepler's cubic close-packing is equal to the ratio of the volume of a sphere to the volume of a circumscribed rhombic dodecahedron (the reciprocal of the cuboctahedron), namely, —
074048
(see, eg., Coxeter 1961, pp. 407-408). W. Barlow (1883) described an equally dense packing in which each sphere still touches twelve others, although the centers do not form a lattice. For this hexagonal close-packing the Dirichlet region is a trapezo-rhombic dodecahedron, which may be constructed by Cut-
ting a rhombic dodecahedron into equal halves by a plane perpendicular to six parallel edges and then sticking the two halves together again after turning one of them through 60° (Fejes T6th 1953, p. 173). 4. Schiafli. Schläfli (1855) studied polytopes in Euclidean n-space and in spherical (n — 1)-space, extending spherical trigonometry from n = 3 to n > 3. He found that an (n — 1)-sphere of radius R in Euclidean n-space has (n — 1)dimensional content (or "surface") and n-dimensional content (or
______________
EQUAL NONOVERLAPPING SPHERES
"volume") JIR*, where —
2K*/t
—
r(f + 1)
—
so that S1 = nj, = S1 = 2,
J1 =
2,
=
2n, S3 = 4r, S4
f2 = ir, f, =
4 J4
23r1,
=
=n8,
S5 =
=
, J3
8
,f,
=
,rt —i—,
He defined (on p. 177) a remarkable function F,1(a), in terms of which a regular simplex of dihedral angle 2a in such a spherical space (with R = 1) has (n — 1)-dimensional content S,F,,(a). When a
= ,r/4, the bounding hyperplanes of the simplex are mutually orthogonal, so that the content is therefore = 1/n!.
When a = n/3, the spherical simplex corresponds by central projection to one of the n + 1 cells of a regular Euclidean simplex; therefore its content is + 1), and
Fn(f) = 2t/(n + 1)!.
(4.3)
The boundary of the spherical simplex decomposes the spherical space into two regions: an "inside" whose dihedral angles are 2a and an "out-ide" whose dihedral angles are 2ir —
2a;
therefore
2'n! F,(a) +
— a)
= 1,
that is, F,,(a) +
(4.4)
— a) = 2k/n!.
In particular, (4.41)
When n =
3,
area 6a —
F3(a) =
(4.5)
When n =
the "simplex" is a spherical triangle with angles 2a, 2a, 2a and therefore, since S3 = 4,r,
2,
ir
—
3
we merely have a circular arc of length 2a; therefore, since
H. S. M. COXETER
56
S1=2ir, (4.6)
Since a regular Euclidean simplex has dihedral angle arcsec n in n dimensions, or arcsec (n — 1) in n — 1 dimensions, and since an infinitesimal spherical simplex is Euclidean, we deduce
a
for
=0
(4.7)
(n — 1).
= ÷ arcsec One of his most brilliant discoveries (Schläfli 1855, pp. 167, 168) is that dF,.(a) =
where
is given by
2a —2. Thus the Schlafli function can be defined recursively by the formula =
sec
(4.8)
ir
sec
sec2P=sec28—2,
(uUsc(.—lflhI
with the initial conditions Fo(a) = F,(a) = 1.
(4.9)
This function has been studied from another standpoint by Ruben (1961, p. 262). 5. fohn, Peechi, Gulnand. Various formulae connecting the angles of the general n-dimensional simplex have been discovered by Poincaré, Som-
merville, Höhn, and Peschl. Their equivalence has been established by Guinand (1959), who observes that, in the symbolic or umbral notation (with standing for a,,), Höhn's formulae become
is the average angle at an (n — k)-cell, expressed as a fraction of the whole angle at an (n — k)-flat. When the simplex is where a,, (symbolically
regular, we have
a,,
=
F,,(a),
so that the formula = becomes
,
' —
, ( F,,(a)=(—1)'Z" .,' F,_,(a).
When r is even, FT(a) cancels, leaving
j.
EQUAL NONOVERLAPPING SPHERES
(—2)'
But
F,_1(a) =
57
(r even).
0
when r is odd, we obtain an expression for F,(a) in terms of simpler
functions:
'
F,(a) =
(5.1)
(r odd).
.,'
From such equations with r =
Fi(a), Fa(a),
1, 3, 5, •••, n, where n is odd, we can eliminate obtaining
-,
=
(5.2)
+
—
--•
—
(n Odd),
where the series ends with the term in Fo(a) and the coefficients are the same as in the expansion tanh x = x
+
—
—
(Schläfli 1855, p. 178).
Alternatively, we can deduce (5.2) from Peschl's formula (2B + a)' = (B + a)'
(r even)
(Guinand 1959, p. 59) or —
1)
i-i
j.
F,1(a) =
j-2
0
or
Since even,
=
0
J.
D B1F,_,(a)
(r even).
when a = (arcsec n)12, we have, for this value of a and n =
+
—
—
(a = (arcsec n)/2, n
For instance, (5.4)
F.(—!2
arcsec 4) = —
a 1
F, (.-i- arcsec 4) 2
farcsec4
—
2 5
—
I
15
even).
H. S. M. COXETER
(Schläfli 1855, p. 182). Other special values that are known explicitly are
f,r\
191
1
(Schläfii 1855, p. 181; Coxeter 1935, pp. 18, 19).
Taking (4.2), (4.3), (4.4), and
(4.7) into consideration, we can now assert that F4(a) is elementary for at
FIGuRE 1
least 13 particular values of a. These results provide a convenient check when numerical computation is applied to (4.8) (with n = 4) or, equivalently, to (5.6)
F4(a)
C—l+3.Ir
ydx,
J—1+(a1c,ecS)Fz
where y is related to x by the symmetrical equation
secirx+seciry+2=O. Figure 1 is a sketch of the curve (5.61) with F(a) indicated as an area in
(5.61)
59
EQUAL NONOVERLAPPING SPHERES
a typical case, actually the case F(ic/4) = 1/24. Figure 2 is a sketch of the curve y = F4(x), with the thirteen "elementary" points marked, namely (with the temporary abbreviation = arcsec k): 4s,
a F(a)
6.
Ifs. - 2\
0
3ks
f:
+= 5)
+:
1
2
191
1
24
15
900
3
409 900
— +s.
8
5
15
8
1/12
s.\
'4599
—
I 2
Minkowukl and Blkhfeldt. We shall return to this function, but first 2 3 5 8
8
is
409 900
1
3
191
900
2 15
0 0
T
ir -i
r
211
FIGURE 2
1!. 2
3i12w 5
3
311
4
H. S. M. cOXETER
we must mention one more historic event: Minkowski (1905, p. 247) proved that the densest packing of equal spheres whose centers form a lattice (in whose centers have" Euclidean n-space) consists of spheres of radius integral Cartesian coordinates with an even sum, not only when = 3 but also when n = 4 or 5. Thus the sphere with its center at the origin touches 2n(n — 1) others, whose centers are given by the permutattons of
In
(±1,±1,0,0)
(n=4)
(±1,±1,0,O,O)
(n=5).
other words, the centers of all the spheres are the vertices of the honey.
and "half measure polytopes" whose cells are cross pol; topes comb hrA (Coxeter 1930, p. 365; 1948, p. 155). When n = 4, this uniform honeycomb is regular, since hr4 = and h85 = {3, 3, 4, 3); the spheres are inscribed in the Dirichiet regions of the lattice, that is, in the cells (3, 4, 3) of the dual honeycomb {3, 4, 3, 3).
Such a packing is rigid for all values of n, but is the densest lattice packing only when n = 3, 4, or 5. When n = 6, 7, or 8 (Blichfeldt 1935) the closest packings are such that each sphere touches 72, 126, or 240 others, respectively. The centers of the spheres are the vertices of the uniform honeycombs ,
331 ,
(Coxeter 1930, pp. 379, 393-397; 1948, pp. 201-205). In other words, for spheres
of radius 3/V2,
convenient
coordinates for the centers are nine integers,
mutually congruent modulo 3 and satisfying the equations
x,+x3+x3=x4+x3+x.=x1+x5+x,=0 Xi+X2+X3+X4 +x-t-x6=x7+x4+x,=0
(n=6), (n=7), (n = 8).
Since Barlow's hexagonal close-packing (in Euclidean 3-space) has the same density as cubic close-packing, namely, = 0.74048..-
it is natural to ask whether some other non-lattice packing may have a greater density. According to Rogers (1958, p. 610), "many mathematicians believe,
and all physicists know" that no such denser packing exists in 3 dimensions. But this has never been rigorously proved, and dense non-lattice packings may reasonably be expected when the number of dimensions is sufficiently great.
Leech has observed that in five dimensions, as in three, there is a nonlattice packing that has the same density as the closest lattice packing. In fact, the three-dimensional honeycomb ho4, formed by the centers in cubic close-packing, may be broken up into a sequence of layers, each consisting of tetrahedra hi3 = a3 and octahedra /93 sandwiched between two tessellations
EQUAL NONOVERLAPPING SPHERES
Hexagonal close-packing is derived by shifting the layers until each triangle {3) of {3, 6) belongs to two tetrahedra or to two octahedra, instead of belonging to one tetrahedron and one octahedron. Somewhat similarly, the five-dimensional honeycomb hl5 may be broken up into a sequence of {3, 6).
layers, each consisting of ht0's and p4-pyramids sandwiched between two fourdimensional honeycombs {3, 3,4, 3). Leech's non-lattice packing is derived by and shifting the layers until each cell {3, 3, 4) of {3, 3, 4, 3) belongs to one
or to two pyramids (the two one pyramid, instead of belonging to two halves of a When the lattice requirement has been abandoned, the packing problem in Euclidean space does not differ greatly from the packing problem in spherical space. The latter begins with the problem of packing small circles on the surface of an ordinary sphere. Meschkowski (1960) states this vividly, as follows: On a planet, say, ten inimical dictators govern. How must the residences of these gentlemen be located in order to get as far as possible from one another? 7. Rankin. Consider a regular polytope inscribed in an (n — 1)-sphere of denote the angle subtended at the radius 1 in Euclidean n-space. Let center by an edge (Coxeter 1948, p. 133). Clearly, the N,, vertices are the packed on centers of N0 nonoverlapping (n — 2)-spheres of angular radius the (n — 1).sphere (e.g., when n = 3, spherical caps on an ordinary sphere). regular simplex afr (1 k is), we have If the polytope is a = it — arcsec k and N,, = k + 1 (Coxeter 1948, p. 295). Rankin (1955) proved that this arrangement is the closest of k + 1(11 — 2)-spheres (1 k is). In other words, letting N(ç6) denote the maximum number of such "caps," of that can be packed on the unit (ii — 1)-sphere, we have angular radius
+1
for ir—arcsecn
it.
Davenport and Hajós (1951)-proved that NW') cannot take a value between n + 1 and 2n, the latter value appearing when the polytope is the cross-poly(Coxeter 1948, p. 121). In other words, tope or "n-dimensional octahedron" although
N(f) = 2n, we have (7.3)
N(#)=n+1
for
Thus NW') is known precisely whenever Rankin
ir/4.
For smaller values of
was content to prove an inequality which yields, for instance,
N(it/10) < 258 when n =
4.
8. The propoeed new bound. It is intuitively obvious that n equal (n — 2)spheres are packed as closely as possible when they all touch one another, SO that their centers are the n vertices of a regular simplex of (angular) edge
H. S. M. COXETER
3
In other words, we may reasonably expect that the density of a packing of at least is "caps" of radius 4' can never exceed 24'.
I-, is the sum of the vertex angles of the simplex, expressed as a fraction of the total angle at a point in n — I dimensions, V is the (n — 1)-
where
dimensional content of such an (ii — 2)-sphere, namely,
v= is the (n — 1)-dimensional content of the simplex. The analogous conjecture for a packing in Euclidean space has been justified by Rogers
and
(1958), with the conclusion that the density cannot exceed
2"(n!)'(n +
arcsec
n).
The truth of the original "intuitively obvious" conjecture has been established
for n = 3 (see Figure 3) by Fejes Tóth (1959, p. 311; see also Coxeter 1962). If it holds universally, we can deduce a bound for N(4') by observing that, since the (n — 1)-sphere is packed with N(4') (as — 2)-spheres of content V, the density is
N(4')j_, and therefore
63
EQUAL NONOVERLAPPING SPHERES
Here 2' is the content of a regular (n — 1)-dimensional spherical simplex and a is the sum whose dihedral angle 2a is a certain function of its edge of the n vertex angles, each equal to the content of a regular (n — 2)-dimensional spherical simplex having the same dihedral angle 2a. By (4.1), 2' = a =n
—
1)!
so that aS,1 — 2F,1_I(a)
2'
—
F,,(a)
-
To express a in terms of & a simple procedure has been suggested by Rogers. Let the vertices of the simplex have coordinates
(c + a, a, -- -, a), (a, c + The angle
a, •-
a),
...,
(a,
a,
c
+ a) -
subtended by two of these points at the origin is given by =
2a(c
+ a) + (n — 2)a'
2ac
— —
+
c+2ac+na2'
whence C
2ac + na
+1
The internal angle 2a between two bounding hyperplanes
{c+(n—1)a}x1—a(x2+-—+x,,)=O and
a(x1+--is given by
cos2a= {c+(n —
2ac+nI
+(n— 1)a2
— c2
+(n
whence
sec2a= 2ac =
Thus our final result is
sec
C
+n—1
+ na + n —2. 2
— lX2ac+ na*)
H. S. M. COXETER
64
sec2a =
(8.1)
+ n —2.
This upper bound is attained whenever the pattern of n spheres touching one another can be continued over the whole spherical space, that is, whenever the centers of the spheres coincide with the vertices of a polytope where p ,r/a. In the notation of Coxeter (1948, pp. 160-162; see also Böhm 1959, 1960) we may conveniently take (n — 2, n) = (sect a)/2,
(j,k)=k—j and (
l(n—1—JXJ-fl,n)—1
(j+1,n—l)
n—2—j
—
whence
(j,n) =
2—j),
—(n
(—1,n)
2
secta — (n
—
1)
(1,n) cos 2a
I —(n—2)cos2a sec
sec 2a — (n
—
2),
as before. For instance, in four dimensions, the regular 600-cell {3, 3, 5) has a = it/S and = 'rhO, so that — —
's.10)
fir\
(2
1
— —
1
—'
The remaining four-dimensional results are epitomized in Figure 4, which shows the smooth curve obtained by plotting the upper bound 2F3(a) F4(a)
for
qS
(it — arcsec
,
sec2a
3)/2, and also the "step function" which expresses the
exact value of for every qS ir/4. For convenience, the accompanying table gives the values of qS and a in degrees instead of radians, so that (4.5) yields 2F;(a) = (a — 30)145:
65
EQUAL NONOVERLAPPING SPHERES
a 2F3(a)
F4(a)
52*
54
18
45
52*
54
36
45
60
72
90
108
3
3
14 15
4 3
26
15 1
1
2
1
3
409 900
8
900
840 — 191
4
1560 409
15 4
212
120
8
5
120 2
15
45
18
135
144
7
38 15
15
5 8
599 900
56 15
2280 599
780
84°
15 14
13 12
It 10 9 8 7
6 5 4
3
2
0 60
12°
180
24°
30°
36°
42°
48°
540
60°
660
72°
90°
96°
FIGURE 4
With only two exceptions (one repeated), every entry in this table is exact. The exceptions are 54j and 52*, which are the approximate numbers of degrees in (arcsec(—3))/2 and (arcsec (—4))/2, respectively. Since F4(a) = 0 when = 0, N(0) 00: the curve for
has the vertical
H. S. M. COXETER
axis as an asymptote. The point (18, 120) is too high to be conveniently shown. The "white" marked points are N(45) = 8, N(521) =
5,
N(54j) =
4,
N(60) = 3, N(90) = 2.
The first three of these, along with N(18) 120, are the only places where and N(#) exactly agree. The "black" marked points help us to plot because, although they have no geometric significance, the curve for they indicate further known cases in which the expression 2F3(a)/F4(a) takes rational values. The lower part of this curve arises because the relation sec2a = +2 makes each value of 4 yield two supplementary values for a, and hence two values for 2F,(a)/F4(a), the larger of which serves as a bound for N(4) while Since the smaller remains in the interval from 3.733 to 3.815 for all < 4
irF4(a) \
= is zero when curve occurs where 9. The cue when
/ sec 2a tan 2cr
ir
the minimum on this nearly straight portion of the = 2r/3.733, which is about 480.
= ir/6.
In any number of dimensions, the centers of three
mutually tangent equal spheres form an equilateral triangle.
Hence the problem indicated in the title of this paper amounts to finding an upper bound for N(ir/6). By (8.1), such an upper bound (attained only when n = 1 or 2) is 2
(9.1)
arcsec arcsec
It is interesting to see how this works out for small values of n. When n = 2, the value
=6 is
attained by the six circles that can obviously touch one of the same size. When n = 3, the bound 2F2(J_ arcsec 3) 2
Fa(+ arcsec 3)
=
6 3 — rfarcsec 3
=
13.39733257
(Coxeter 1958, p. 757), like Rankin's
N=
+ 8 = 16.48528...,
is not strict enough to disprove Gregory's conjecture that 13 spheres in
67
EQUAL NONOVERLAPPING SPHERES
ordinary space could touch one equal sphere. When n = 4, we use (5.3) to compute
\
1
=
\
/1
1
2
=
—
1 farcsec4
2 —
Since, by (4.5),
/1
\= arcsec 4)
arcsec4
1
— —,
it follows that -
/1
2
arcsec 4)
arcsec 4)\
= 5(1 +
\
1
5—2ir/arcsec4 )
= 26.44009910
Though this number is less than Rankin's —
1Y4
10—
3r
—
.
it leaves open the question whether a 3-sphere in Euclidean 4-space can touch only 24 others (as in the closest lattice packing) or 25 or 26. 2a) 10. Leech'8 computation. When n > 4, it is convenient to write for F.(a), so that =
arcsec
x).
In this notation, the formulae (4.6), (4.7), (4.8), (5.2), (5.3) become
f2(x)= arcsec x
f,,(n—1)=0,
,
ir
IA)
—
=
xVx
—1
4-ia .3(x) + —
—
I
(n odd),
7(1) +
—
(n)
(n
even),
and the function (9.1) is simply
Leech helpfully undertook to evaluate this function for by tabulating arcsec(x —
ii
8.
He began
2)
—1
to twelve decimal places for x =
3.5(0.1)8.2,
and integrated by the trapezium
II. S. M. COXF.TER
68
rule with central difference correction at the ends of the ranges. From f4(x) he deduced
=
+
—
and
f.(6) = +1446)
17
—
+
He obtained fe(x) by a similar integration of •
f4(X — 2)
zrxVx'
—1
Finally, he deduced
f,(x) =
17
— 114(X) +
—
and
= +f6(8)
62
+
—
The results are as follows: =
f(6) = 0.00003 02840 12, /6(7) = 0.00010 31285 78, 15(8) = 0.00018 73637 63, /7(7) = 0.00000 14072 22, /7(8) = 0.00000 64990 72, = 0.00000 00531 355,
31255 82, 14(5) = 0.01248 11393 80, /4(6) = 0.01686 59385 17, f4(7) 0.02014 88228 26, 0.02268 05970 96, = 0.00051 25449 97, f3(6) = 0.00129 93981 97, 0.00652
2 13(4)/14(4) = 2
26.440...
= 48.702
2
fs(6)/fG(6) =
85.814...
2 16(7)1/7(7) = 146.570 2
f7(8)/fe(8) = 244.622...
Thus the maximum number of spheres that can touch another of the same
size (in Euclidean
is
N1=2,
where N2
=6, 26, 48, 85,
126
N1
240
N6
146, 244.
N3=12,
EQUAL NONOVERLAPPING SPHERES
69
Notice the remarkably small difference between the upper and lower bounds for Ne. This closeness seems to be a manifestation of the extraordinary "near-regularity" of the honeycomb 5k,, whose vertices are the centers of the spheres in the lattice packing. Any simplicial cell of the honeycomb indicates a set of nine spheres, perfectly packed. Of every 137 cells, 128 are simplexes and only 9 are cross polytopes (Coxeter 1948, p. 204). The same honeycomb appears again in the theory of Cayley numbers (Coxeter 1946, p. 571) and in the related theory of the simple Lie group E8 (Coxeter 1951, pp. 412-414, 420— 426). 11. Spherea In non-Euclidean spacee. The problem that plays the title role in this investigation remains significant when the equal spheres are packed
in a non-Euclidean space (spherical, elliptic, or hyperbolic).
The 3-dimensional
case has already been considered by Fejes Toth (1954), who uses spheres of radius 1 in a space of curvature K. The number of such unit spheres that can touch another unit sphere is evidently is half the angle where of the equilateral triangle of side 2 whose vertices are the centers of three mutually tangent spheres. By the familiar non-Euclideari formula
sinA= sinai/K sin
for the angle A of a right-angled triangle ABC, we have
_sini./R'_
-
—
sin 2i/TI?'
1
2cosifK'
whence + 1.
sec 2çS = sec
By (8.1), an upper bound for the number of spheres that can touch one is 2F,,-,(a)
where a is given by
sec2a=sec2VR'+n—l
-
In the special case when n = 3, we see from (4.5) and (4.6) that 12a 6a —
By (11.2), we have sec 2a —
1 + tant a 1 — tant a
— 1
1
=
+ tant
— 1 —
tan2
sec 2VR'
VR' +
cot2 a + Since
'
+ 1, whence tan2 a
—
tan2
a—
1 —
2.
1
1 —
tan2
VTK'
H. S. M. COXETER
a=
N—26'
this is equivalent to the formula of Fejes Toth (1954, p. 162). 12. An asymptotic bound. How fast does the number of spheres increase when the number of dimensions tends to infinity? According to Rogers (1961), when the number =
sec
2a
n+1
(cf. (11.2)) is bounded,
/1 + nb I I' 2e \.IS y 2 n! ett' k.irnb) When we apply this asymptotic formula to (8.1), we have b ' sec — I in Sec 2qS in the numerator. Thus the asymptotic the denominator, and b upper bound for is
( \n — ii
21
(12 1
\312
\. e
I
This excels, by the factor 2/e, the bound cos
n312
sin'_I
of Rankin (1955, p. 139; our is his a). Setting = in (10.1), we deduce, for the bound named in the title of this paper, the asymptotic expression
REFERENCES
W. Barlow 1883. Probable
nature of the internal symmetry of crystals, Nature 29,
186 188.
C. Bender 1874. Bestimmung der grössten Anzahl gleich grosser Kugein, welehe sick auf eine Ku gel von demselben Radius, wie die übrigen, aufiegen lassen, Grunert Arch.
56. 302 313.
II. E. Illichfeldt 1935. The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39, 1-15. 1959. Simplexinhalt in konstanter Krummung betiebiger Dimen j. nion, J. Reine Angew. Math. 202, 16-51. !nhaltsmessung im
konstanter Krummung, Arch. Math. 11, 298-
II. S. M. Coxeter 1930. The pot ytopes with Trans. Roy. Soc. 1.ondon, Ser. A 229, 329 -425. 1935.
The
vertex figures (1). Philos.
functions of Schtäfii and Lobatschefsky, Quart.
Ser. 6, 13-29. 1946.
integral C'ayley numbers. 1)ukcMath. J. 13,
561 -578.
J.
Math. Oxford
EQUAL NONOVERLAPPING SPHERES
71
1948.
Regular polytopes, Methuen, London; 2nd ed.. Macmillan, New York (to
1951. 1954.
Extreme forms, Canad. J. Math. 3, 391-441.
appear).
Arrangements of equal spheres in non-Euclidean spaces, Acta Math.
Acad. Sci. Hungar. 5, 263-274.
Close-packing and froth, Illinois J. Math. 2, 746-758. Introduction to geometry, Wiley, New York. 1962. The problem of packing a number of equal nonoverlapping circles on a sphere, Trans. New York Acad. Sd. (2) 24, 320-331. H. Davenport and G. Hajós 1951. Aufgabe 35, Mat. Lapok 2, 68. G. L. Dirichlet 1850. (iber die Reduktion der positiven quadratiachen Formen mit drei unbestimmten ganzen Zahlen, J. Reine Angew. Math. 40, 209-227. L. Fejes Tóth 1953. Lagerungen in der Ebene, auf der Ku gel und im Raum, Springer, 1958.
1961.
Berlin.
On close-packings of spheres in spaces of constant curvature, Pub!.
1954.
Math. Debrecen 3, 158-167. 1959. Kugelunterdeckungen mind Kugelüberdeckungen in
konstanter Krumi'nung, Arch. Math. 10, 303-313. S. Günter 1875. Em stereometrieches Problem, Grunert Arch. 57, 209-215. A. P. Guinand 1959. A note on the angles in an n-dimensional stm pIes, Proc. Glasgow Math. Assoc. 4, 58-61. S. Hales 1727.
Vegetable staticks, London.
J. Kepler 1911. Dc nive sexangula, Gesammelte Werke Rd. 4, pp. 259-280, Beck, Munich.
J. Leech 1956. The problem of the thirteen spheres, Math. Gaz. 40, 22-23. H. Meschkowski 1960. UngelUste und unlösbare Problem. der Geometne, Vieweg, Braunschweig. H. Minkowskj 1905. für arithmetische Aquivalenz. J. Reine
Angew. Math. 129, 220-274. R. A, Rankin 1955. The closest packing of spherical cap. in n dimensions, Proc. Glasgow Math. Assoc. 2, 139-144. C. A. Rogers 1958. The packing of equal spheres, Proc. London Math. Soc. (3) 8, 609-620. 1961.
An asymptotic expansion for certain &hldfli functions, J. London
Math. Soc. 36, 78-80.
H. Ruben 1961. A multidimensional generalization of the inverse sine function, Quart. J. Math. Oxford Ser. (2) 12, 257-264. L. Schlafli 1855. Rt1duction d'une integrale multiple, qui corn prend i'arc Se cercie et l'aire du triangle sphrrique comm. Ca. particuliers, Gesammelte mathematische Abhandlungen, Bd. 2, pp. 164-190. (See also pp. 219-258.) IC. Schutte and B. L. van der Waerden 1953. Des Problem dcv dreizehn Kugein, Math.
Ann. 125,
325-334.
G. Voronoi 1908.
134,
Recherches sur tee paralléloèdres primitives, J. Reine Angew. Math.
198-287.
UNIVERSITY OF TORONTO
ROTUNDITY
Ii F. CUDIA The theory of Finsler spaces and, I. Introduction and general more recently, that of spaces which are locally Banach (of arbitrary dimension) has been developed considerably in the last decade. The theory of curvature
in spaces of the latter type has been considered in Ewald and Kelly [1] and in a paper by Lorch [IL For further references to such spaces see Segal IlL Eells [1], Laugwitz [1], and Rund (1).
However in the article (2) H. Busemann observes, "Clearly the step from no geometry to euclidean geometry is incomparably much wider than that from euclidean geometry to Riemannian geometry. The volume problem makes
it more than probable that an analogous situation exists for Finsler spaces. Therefore the study of Minkowskian geometry ought to be the first and main step, the passage from there to general Fin,sler spaces will be the second and simpler step." While there is no obvious analogue to volume and area in a general Banach space, nevertheless it is in the spirit of this quotation that this survey of metric geometry in normed spaces is conceived, namely, as a necessary and important preliminary to the study of infinite dimensional Finsler spaces. In the paper [3) Busemann uses the technique of replacing the unit sphere of a Minkowski space by a more convenient convex body. The technique is useful in general Banach spaces and leads to isomorphism results which are results serve to enlarge the uses of geometry considered in § V.
in the theory of Banach spaces and for that reason are as important as the investigation of the various geometric properties that the norm itself may possess. Although rotundity and smoothness concepts can be defined in general locally convex spaces, the proper domain of these ideas is in a normed space where the theory of linear homeomorphisms can be used to renorm the space without changing the topology so that the geometric theory of the new norm
can be used in the study of the space. Lying within the general framework of infinite dimensional Finsler spaces, then, the purpose of this report is to survey rotundity and smoothness concepts and the theory of linear homeomorphisms in normed linear spaces and especially to provide an up-to-date bibliography of papers which are concerned
with these ideas. The study of non-linear homeomorphisms in infinite dimensional normed spaces is not covered either in the survey or in the bibliography. Some applications to approximation theory are given in § VII but other applications to ergodic theory, fixed point theory, and extreme points are omitted. Aside from the notions of local uniform smoothness and midpoint local uniform smoothness none of the geometric properties considered is original. 73
D. F. CUDJA
74
The pressures of time forced the omission of a detailed study of these two properties. See however Cudia (11 for a discussion of related matters.
In preparing this report the author had the benefit of many stimulating conversations with Dr. M. M. Day whose grasp of the literature on the subject both as to location and importance gave this report any completeness it possesses. The notation and many of the results have been taken directly from the second section of Chapter VII of the book, Normed linear Any misinterpretation of these results or the quoted results of other authors however is entirely due to the present writer. A subset of a normed linear space is called a conrex body provided it is bounded, closed, convex, and has a nonempty interior. A convex body is said to be rotund provided every hyperplane of support of the body has at most one point of contact with the body. It is said to be smooth provided it admits through each boundary point only one supporting hyperplane. The definitions have been phrased to exhibit their dual character. By defining the polar body §81 or Bon of a given convex body in a Euclidean space as in Minkowski the rotundity nesen and Fenchel ti, p. 281 we can observe a duality between or smoothness of a convex body and its polar body. A nornied linear space is said to be smooth or rotund if its unit ball,
U= {x:IIxll 1) is smooth or rotund. For finite dimensional normed spaces the duality is complete because the conjugate space is then congruent to the space itself, under a norm such that the unit balls are mutually polar. A normed linear space B is said to be uni/orrnly rotund, Clarkson Ii), if there exists a function 8 such that 0 < 8(8) if 0 < then llyH = and llx—yU
2 and such that if II
=
1
X
I
—
ii
An
equivalent condition for the rotundity of a normed space is that the
boundary of the Unit ball contain no line segment. The compactness of the surface of the unit ball in a finite dimensional normed space implies the equivalence of uniform rotundity and rotundity for such spaces. Footnote 13), that the It has been observed, Alaoglu and Birkhoff dition that the unit ball have two support planes at the same point is equivalent to the condition that the unit sphere in the conjugate space contain a straight line segment. While this is not true in all Banach spaces, there is complete duality of smoothness and rotundity for reflexive spaces. This footnote is the first observation of the duality between rotundity and ness in infinite dimensional spaces that the author has been able to find. Klee [3j extends the concepts of points of smoothness and points of rotundity
to an arbitrary convex subset of a Hausdorff linear space F in the following manner. We work In terms of a linear space F of linear functionals on F. By an F-hyperplane is meant a set of the form I '(r) = {x:f(x) = r, XE E)
75
ROTUNDITY
for some nonzero fe F and some real number r. The set C is supported at a point ft by the hyperplane H = f'(r) if p C A H, C lies on one side of H, that is, supf(C) r or inff(C) r, and C is not contained in H; p is then said to be an F-supPort point of C, otherwise an F-nonsupport point. A point p of C is said to be a point of F-s,noothness provided all the F-hyperplanes which support C at p have the same intersection with the affine extention of C, a point of F-rotundity provided each F-hyperplane which supports C at ft intersects C only at p, and an F-exftosed point provided C is supported at p by an F-hyperplane H which intersects C only at p. The convex set C is said to be F-smooth provided each of its points is a point of F-smoothness, and to be F-rotund provided each of its points is a point of F-rotundity. There are three especially important choices for F: the space Es of all continuous linear functionals on E, the space of all linear functionals whose restriction to C is continuous, and the space of all linear functionals on E. FOf course when the convex set C has an interior point, rotundity coincide with the previous definitions for each of the above three choices for F. The geometry of Banach algebras has been investigated by Kadison Ill and 'F he latter show that irrespective of the norm Bohnenblust and Karlin the hyperplanes of support at the unit element on the unit sphere of a Banach algebra are inverse images of a total set of continuous linear functionals.
This is equivalent to utter "unsmoothness" at the identity element on the surface of the unit halt; more precisely, to the condition that the intersection of the unit ball with every two-dimensional linear subspace through identity has a corner at the identity. in a C5 algebra the hyperplanes of support at the unit element, i.e., the set of linear functionalsfwhich satisfy the conditions f(zi) = 1, and if ii = I is shown to coincide with the positive linear functionats normalized at u. Thus it is geometrically evident that a linear mapping of norm one of a C5 algebra into another is order preserving if it transforms the unit element into the unit element. Smulian 141 considers differentiability of norms in Banach algebras. The metric geometry (>1 semiordered Banach spaces was first studied by Smulian 131 who obtained a sufficient condition for reflexivity in terms of uniform Fréchet differentiability of the norm on a subset of the positive cone. More recently, Ando 12I, Anieniiya, Ando, and Sasaki Luxemburg 11), Mimes Ill, Nakano (ii, and Yamamuro ii; 21, have studied geometrical questions in modulared vector lattices. A modulared vector lattice is a boundedly complete vector lattice (Day 17, p. 96i) L on which a functional called a nw(Iulur is defined. The least upper bound of elements x and y in L is de-
noted by x V y and the greatest lower bound by x A of x is defined by lxi = (xv 0)— (x A 0).
.
The absolute value
With this notation, the modular m must satisfy the following conditions: (1) (2)
0 ';z(x) if
=
0
for all xeL; 0, then x
for all
0;
D. F. CUDIA
76
(3) (4)
for any xc L there exists a > 0 such that ,n(ax) < oo; for every xe L, ,n(Ex) is a convex function of E > 0, that is, a, >
0
implies 2
1 X) T{m(ax) +
m(x) m(y); implies m(x+y)=m(x)+m(y); monotonically increasing to y implies
(6) xAy=0 (7)
0
XA
m(y) = sup m(xA). A
For any p L, [p] denotes the projection operator associated with p,
Ep](x)=
for each
0.
VXA njpl
For any x€L,
[p](x) = [P][(x v 0) — (—x V 0)] = [p](x
V
0)— [p](—x v 0).
a positive linear operator in L. If P and q are in L then [N and only if for any x 0 in L we have [P1 is
[q) if
[qI(x).
Two norms can be associated with each modular m: tIxII
and
iIIxlIl = mt called respectively the firs! norm and the second (or sometimes ,nodular) norm of m. The modular norm is the Minkowski functional of the set {x: m(x) 1). The spaces (L, II . II) and (L, III'S flJ) can be shown to be (4B)-lattices in the terminology of Day [7, p. 98]. (L, II II) and (L, Ill) are isomorphic in the
sense of Banach [1, p. 180]. In fact, Nakano [1, § 40] proves that lll.*11l
IlxII 2111x111
of the metric geometry in modulared vector lattices cited above is directed toward finding conditions on the modular in order that one or the other of its associated norms be smooth, rotund, or uniformly rotund. We shall mention a few of these results in the appropriate sections of this survey. Most
II. of rotundity and smoothness properties; notation. In this section we list some generalizations and uniformizations of rotundity and smoothness. The abbreviations preceding the conditions will facilitate the statement of theorems in the sequel. In this section B will be a Banach space, U its unit ball; S, the boundary of U, defined by
ROTUNDITY
S=(x:lIxll =1); U', the unit ball of B*, and S' the boundary of U'. (R): B is rotund (R) if and only if S contains no line segments. (MLUR): B is midpoint locally uniformly rotund (MLUR) if and only if given £ > 0 and x0 in S there exists ö(i, x0) > 0 such-that
ttX0_ X±Yll8 whenever Jix—yll i and x,y are in S. (LUR): Lovaglia [1). B is locally uniformly rotund (LUR) if and only if given e > 0 and x0 in S there exists 8(e, x0) > 0 such that
'II
ll8
whenever lix— xolI i and xis in S. (UR): Clarkson [1]. B is uniformly rotund (UR) if and only if given i > 0, there exists 8(1) > 0 such that
whenever ljx—yIi > i, and x andy are inS. (kR): Smulian used (2R) in [4]; Fan and Glicksberg [1). B is k rotund (kR) if and only if every sequence (xe) from B such that
TE"; =1 11
urn
a convergent sequence. If every such sequence is weakly convergent then B is said to be (WkR), weakly k rotund. (WLUR): Lovaglia [1]. B is weakly locally uniformly rotund (WLUR) if is
and only if for each i with 0 0
ii F. CUDIA
78
and f is in S'. (PLUR): Fan and Glicksberg
[21. B is Point locally unifor,nly rotund (PLUR) if and only if whenever a sequence (xe) in B with
=1 has no weak cluster point of norm < 1, and if, for some Xo ill S
then
=0. (S):
B is smooth (S) if and only if at every point of S there is only one
supporting hyperplane of U. (MLUS): B is midpoint locally uniformly smooth (MLUS) if and only if for each e > 0 and each x0 in S there is a 8(e, x0) such that
lko_
X+Yll 0
—yfl II
whenever (lxo — xfI (US):
for each
x0) and y is in S.
Day 141 and Nakano 12). B is uniformly e > 0 there is 8(e) > 0 such that
smooth (US) if and only if
i whenever Jix—yll (G):
and x andy are inS.
B is said to be Gateaux differentiable (G) if and only if
G(x,y)=Iim IIx+tylj—iIxIl £
exists for each x and y in S. (UG): Smulian [2]. B is said to be uniformly Gateaux differentiable (UG) if and only if the limit in (G) is approached for each y in S uniformly as x varies over S. (F): B is said to be Fréchet differentiable (F) if and only if the limit in (G) is approached for each x in S uniformly as y varies over S. (UF): B is said to be uniformly Fréchet differentiable (UF) if and only if the limit in (G) is approached uniformly as x and y vary simultaneously over S.
(A): B is (A) if and only if when a sequence
in B converges weakly
79
ROTUNDITY
to x0 and
limlix.lI = lix, II, then
—x0(l =0. Fan and Glicksberg [2]. B is (H) if and only if B is (A) and B is (R). (K): B is (K) if and only if when K is a convex set in B then D(Kri tU), the diameter of K n tU, tends to zero as I decreases toward the distance from K to 0. (D): Smulian [2). B is (D) if and only if for any f in S', when we define (H):
E(f,8)=
1—8) n U,
we have D(E(f, 8)) tending to zero as 8 decreases toward 0. (D*): Smulian [21. is (D*) if and only if for any x in S with E(x, 8) =
{f:f(x)
I
—
8)
fl
U'
we have D(E(x, 8)) tending to zero as 8 decreases towards 0. (DL): (D localized). The conclusion of (D) holds for each f in S' which attains its supremum in U at some point of S. (W*UR): Smulian [21. B* is weak* uniformly rotund(W*UR) if and only if for each £ > 0 and each x0 in S there is a positive 8(e, x,) such that if f' are in S' then the inequality and f1
11>1
implies that —
I
Smulian (21.
(WUR):
f2(x0) I
0, there is a 8 > 0 such that if x is in S and whenever Ix — ILvil
v3 then
lix +-yli + II x —yII
2 + iIyIl
ROTUNDITY
B is (US) if and only if B is (UF). Clearly B is (UF) implies B is (UG) or B is (F) either of which implies B is (G). Klee [5] shows that B is (UG) implies that B is (0). It is also true that B is (0) implies B is (G). Smulian 121 indicates that B is (UG) is equivalent to B is (W*UR) and that B* is (UG) is equivalent to B is (WUR). B is (LUR) implies B is (H) and the converse is false according to Ander• son 11), where it is also shown that if B is reflexive and (H) then B is (MLUR). Lovaglia f 1] shows that B is (LUR) if and only if B is (DL) and B is (WLUR).
Fan and Glicksberg [21 note that a condition equivalent to "B is (K)" is obtained by requiring that the defining condition be satisfied only for closed hyperplanes or only for closed half-spaces. Day [7, p. 113] indicates that B is (kR) implies B is (K); and that B is (K) if and only if B is (D). B is (D) if and only if B is (DL) and every supporting hyperplane of U meets S. B is (DL) implies B is (H) according to Day p. 113]. Also, Day (1, p. 113], B is (LUR) does not imply B is (D) but B is (LUR) does imply B is (DL).
Fan and Glicksberg [2] show that B is (PLUR) implies that B is (H) and show that B is reflexive and (PLUR) if and only if B is (K) if and only if B is reflexive and (H). Whence B is (K) implies B is (MLUR) and B is reflexive and (PLUR) implies B is (MLUR). Yamamuro [11 and Ando 12] study conditions on the modular necessary and
sufficient to insure that the modulared vector lattice L be (R) under its first or second norm. L is said to be non-atomic provided for any 0 < z L there exist x,y in L with x> 0, y >0 and x A y = 0 such that Ando 121 proves that in a non-atomic modulared vector lattice L with modular in, (L, II II) is (R) if and only if m is strictly convex and m is infinitely inor — = = creasing, i.e, for every x * 0, = if r < (L, III Ill) is CR) if and only if m is strictly convex and -
-
infm(
x
\UlxIIIJ
(L.
II
-
II) is (S) if and only if in is even, infinitely increasing, and finite, i.e.,
for any x in L.
(L, III 'Ill) is (S) if and only if m is even and finite. m is
uniformly convex if and only if (L, II II) is (UR) if and only if (L, III . III) is
(UR). m is uniformly even if and only if (L, II
II)
is (US) if and only if
CL, III. III) is (US).
Related notions of flatness and uniform flatness are treated in Amemiya, Ando, and Sasaki [1]. IV. Product 8paces, quotient spaces, and subapaces. In this section questions
concerning the extent to which the geometry of the factors influences the geometry of the product of Banach spaces are considered. The remainder of the section deals with the extent to which the geometry of the space is in-
D. F. CUDIA
herited by its quotient spaces and subspaces. Let S be an index set and let X be a Banach space of real-valued functions on S. If for each s in S a normed space N is given, let PEN. be the space of all those functions x on S such that: (1) x, is an element of N. for every s in S, and is the real-valued function defined by (ii) if
= llx,lI for each $ in S, then is in X. We define the norm of x in PEN,, by (I xii = ii If X satisfies the condition that whenever is in X and E(s)
—
for all s
then
is in X and II'?Il
Ii
Eli
PEN. is a Banach space if each of the N, are Banach spaces. We shall by P,N. 1 and Clarkson [II showed that
then
denote
and (UR) if space. Boas 11) generalizes this to
is
a finite dimensional rotund Banach or PP(L,[O, 1)), and to PAN, where
or all of the N, are LQ[O, 1), and p X is L,[O, 1] and all of the N. are and q are greater than 1. The last two product spaces are the spaces of all or Lq[O, 1], respectively, which are functions on 10, 1] with values in integrable in the sense of Bochner [11. Day [2] showed that P2(lg;(lc)) is (UR) if there exist 1 <m M < :n such that m
M
for all I in w, and Day El) shows that otherwise is not even isomorphic to a (UR) space. Finally, Day [3] extends all of these results in showing that is (UR) if and only if X is (UR) and the spaces N. have a common modulus of rotundity. Since (US) is dual to (UR) (see § VI) a similar statement holds for the uniformly smooth product of uniformly smooth Banach spaces. Lovaglia [1] proved that PAB, is (LUR) if Xand all B, are locally uniformly rotund Banach spaces. Fan and Glicksherg [1) show that when I, > 1, and k
is an integer greater than 1, P,B, is (kR) if each of the B. are k rotund 1, P,B, is (MLUR) if each B, is (MLUR) and Day [5] shows that is (R) if X and each B, is (R) and proves a general theorem which gives that for p> 1, P,B, is (S) if each B. is (S). Anderson [1] shows that for p 1, is (A) if each B. is (A) and this combined with the result of Day [5] shows that for p> 1, is (H) if each B, is (H). For a closed linear subspace L of a Banach space B the quotient space B/L is a Banach space whose elements are the translates of L with the norm
Banach spaces. Anderson Eli shows that for p>
ROTUNDITY
infjlx+yII IIx+ LII = VEL [41, Day showed that B is (UR) [(US)] if and only if there is a common modulus of rotundity [smoothness] for all the quotient spaces of B and if and only if there is a common modulus of rotundity [smoothness] for all the two dimensional quotient spaces of B. Day 17, p. 114] observes that a sufficient condition for rotundity [smoothness] of B is the rotundity [smoothnessj of every two dimensional quotient space of B. It is further shown that this is not a necessary condition for rotundity and Day conjectured the same for smoothness. Klee [3] substantiates this conjecture and shows that if L is a reflexive subspace of B then smoothness or rotundity of B is transmitted to the quotient space B/L and that in general no more can be expected. Thus rotundity [smoothness] of every two dimensional subspace of B is necessary but not sufficient for the rotundity [smoothness] of every two In
dimensional quotient space of B. However, in the case that the two dimensional subspaces have a common modulus of rotundity [smoothness] these conditions are equivalent since then B is (UR) [(US)] and hence reflexive. Clearly, if B
is (UR) [(US)] then the subspaces of B have a common modulus of rotundity [smoothness]. V. Isomorphism. Banach [1, p. 180] called two normed spaces isomorphic if they are homeomorphic under a linear function. The general problem considered in this section is: When is a Banach space isomorphic to another Banach space which has more pleasant geometric properties than the first? This problem is equivalent to that of reforming the space without changing the topology in such a way that the new norm has some geometric properties not possessed by the first. Since the theory is still fragmentary this section is divided into two parts; general results, and results valid only for special spaces. The more complete results available for modulared linear lattices are inserted in both parts wherever it seems appropriate. Lower case letters are used as in Day [51, i.e., the phrase, "a normed space B is (q)," means that B is isomorphic to a normed space having property (Q). In the cases where B is isomorphic to a normed space which is both (P) and (Q) we shall write that B is (pq). Clarkson [I] proved that any separable Banach space is (r). Klee [2] showed that every separable reflexive Banach space is (r) and (s). Day [5] improved on both of these by showing that any separable Banach space is (rs). Finally, Klee [3] showed that any separable Banach space B is (rs) with 11* also (R) under the new conjugate norm. This result cannot be strengthened to also obtain smoothness for B * for m(h) = is not (s). In [4] James showed the existence of Banach spaces with any specified number of separable conjugate spaces. Note that if B is a Banach space whose nth conjugate space B " is separable and if n is even then B admits a norm with respect to which B is (R) and (5) and each of the even numbered conjugate spaces from B = B'°' to B IR is (R) and (S) in the norm induced on B" by
In the same manner, the rotund conjugate norm of that in B induces a rotund norm in each of the odd number conjugate spaces from B' = B* to
D. F. CIJDIA
B A similar situation prevails when n is an odd integer. Of course the may not be an nth conjugate norm. James [3] gives an new norm in example of this situation when n 4. Day [5] raised the following question: Is there a nonreflexive, nonseparable
space B such that B is (rs)? Wada (1] answered this in the affirmative by product (p> 1) of C0, the set of all sequences of real taking for B the numbers which converge to zero, and lp(IM) (p> 1). Still unsettled is the question of the existence of a reflexive space which is not (r) or of one that is not (s). Isomorphism questions in moduLared vector lattices can be given almost complete answers. Two modulars in1, ,n2 on a boundedly complete vector
lattice L are said to be equivalent if their modular norms are isomorphic. Every modular is equivalent to one which is infinitely increasing, and continuous, I.e., for any X€ L if m(x) = oo then sup
IriSki;
If the original modular is strictly convex [even], the new one can be also. Suppose that (L, in) is a modulared vector lattice which is complete under the modular norm and that 0 x. monotonically increasing with supA x x
that (L, (II is (r) if and only if (L, III ifi) does not contain a closed vector sublattice linearly isometric and lattice isomorphic to with a continuous norm It can be proven that a complete (x.. going monotonically to 0 implies U II goes monotonically to 0) is (r). According to a result of James [2, Theorem 1) and a result of Lovaglia [I, Theorem 3.1]. if a reflexive Banach space B has a basis then B is (lur) and hence (plur). By the results of III, this last conclusion may be replaced by (k), (d), (plur), or (h). Lovaglia also proves that if B has an unconditional basis and does not have a closed subspace linearly homeomorphic with co(Ie) then B is (lur) 1(wlur)1. Fan and Glicksberg L21 prove that if a B is (H) and Bk is separable then B is (lur). An immediate consequence is that if B is any separable normed linear space such that B is (K) (or (D)) then B is (lur). Kadec [1] shows that in the space CEO, 1], if for fe C[0, 11, rn i112 1 + sup If(r) —f(s)I 11111 = sup If(t)I + 1112] -
LJO
k
i2
Since every separable Banach space is linearly homeoII II is (H). morphic with a linear subspace of CEO, 1], Kadec 11] obtains the result that every separable Banach space is (h). Anderson (1] uses this to improve the theorem of Fan and Glicksberg stated above to say that if B* is separable then B is (lur), and hence every separable reflexive Banach space is (lur). then
That separability of Bk is not a necessary condition for B to be (lur) evident from an example of Phelps [IJ wherein
is
is shown to be (Jur) but
87
ROTUNDITY
is not separable. However, Kadec [2] has asbeing congruent to serted that every separable Banach space is (lur). Klee [4] shows that if B is a Banach space and B* is separable then B* is (h*), i.e., there is a topologically equivalent norm II . II in B* such that (B*, II U) is (R) and whenever U —' If U, and f,, is weak* convergent to is a sequence in B*,f in B*,
f, then lIfe
—f II —GO.
Fan and Glicksberg [1] show that under the same
hypothesis, B is (w2r). The last two statements are true in particular when B is separable and reflexive. Let A and B be normed linear spaces, and define
k(A,B)= iiriist sup inf uTah Then k(A, B)> 0 if and only if A is isomorphic to some subspace of B. [41
shows
Day
that a necessary condition for isomorphism of B or 1? with a
uniformly rotund space is that urn k(11(n), B) = urn k(111(n), B) =
0.
Further considerations concerning the uniform rotundability of normed spaces and modulared vector lattices and the modularibility of complete (AB)-lattices are given in VII. A fundamental question is under what conditions a vector lattice is modularable. Klee [21 proves that continuous linear images of a reflexive Banach space which is (r) [(s)J are also (r) [(s)]. Day [5] gives a related result whose application is not restricted to reflexive spaces: If B0 is a reflexive smooth
space and T is a linear function from B0 onto a dense subset of B, then B is (s) and B* is (r). We shall apply this smoothability theorem to the spaces c0(!) and L1(p) later on in this section. There are examples of nonreflexive nonseparable spaces which are (r) and (s), and nonreflexive spaces which are (r) but not (s), or neither (r) nor (s). There is no example yet known of a Banach space which is (s) and not (r). Ando (21 has shown that if the modulared vector lattice (L, m) is complete under the modular norm then (L, III III) is (s) if and only if (L, III III) does not contain a closed vector sublattice linearly isometric and lattice isomorphic to m(10) or
Day [5) considers isomorphism problems in product spaces. There it is shown that for p 1 the 1,(I) product of rotundable spaces is rotundable, the product of such spaces is (r), and if B is (r), then the space of continuous functions from K(I) into B is (r). If all B, are (s) and if I is countable or if p > 1, then the 4(I) product of the is (s). Criteria for rotundability and sinoothability in terms of subspaces are obtained by Day [5]. There it is shown that a normed space B is (r) if and only if there is a one-to-one and linear map of B into a normed space which
is (r), B is (s) if B is isomorphic to a subspace of a normed linear space which is (s). A related result is that B is (rs) if B is (s) and there is a oneto-one linear map from B into an (rs) space. Ando [2] shows that in a modulared vector lattice (L, m) which is complete under the modular norm,
1). F. CUDIA
88
III) is (ur and us) if and only if (L, Ii III) contains no closed vector sublattices linearly isometric and lattice isomorphic to C0(fe) or
(L, Ill
In a reflexive space B, an isomorphism of either B or B' determines an isomorphism of the other so that in such spaces all the duality results of § VI can be written in lower case letters. For nonreflexive spaces an isomorphism of B' may exist which is not determined by an isomorphism of B, but Klee [2] has observed that if B' is (r) [(s)J and if the new unit ball is weak' closed, then B is (s) [(r)j. For a pair of spaces B and B' in the nonreflexive case there are examples with B and B' both (rs) or B (rs) and B' (r) but not (s) or B (r) but not (s) and B' neither. There is no example yet known with B (s) and B' not (r). (But recall from § IV that Klee has produced a space where
B is (S) but B' is not (R).) Anderson [1) uses a result given in § Vito prove that if B' is (lur) and if the new unit ball is weak' closed, then B is (f). Other isomorphism results can be generated from those of § VI with the extra hypothesis of weak' closure of the new unit ball. As a further example (Anderson (1]), if B' is (f) then B is (mlur).
As an example of an isomorphism duality result in a modulared vector lattice (L, m), Ando (21 observes that L' is (r) if and only if m is equivalent to an even modular (see § VI). Relations between reflexivity and isomorphism results are given in § Vi!.
In the remainder of this section results on special spaces will be listed. These results will facilitate the application of the theory. All of what follows is in Day (51, Klee [3] or Wada [1]. In the following table, a plus (minus)
means that the space in that row has (does not have) the property in that column; a question mark means that the precise nature is not yet known. This table is the same as that in Day (5] but brought up to date in the light of present knowledge. In what follows and represent countably infinite and uncountable index sets. ço is a finite or a-finite measure sufficiently nontrivial that is not finite-dimensional; &' is a finite or a-finite nonseparable measure; A is a non-afinite measure. B separable cO(!R)
(rs) + ?
(r), (s)
(r), not (s)
(s), not (r)
not (r), not (s).
+ + ÷
m0(1.)
+ +
4(l),/>1
+
L1(i')
?
+ +
+
+
+
L1(A)
L,(ie),P>1 M(c)
+
+
M(A)
C(K(h))
+
ROTUWIMTY
The first examples of spaces not rotundable or smoothable were given in is is not (r). There it was shown that m0(I,,) and hence not (s) and if I is infinite, then m0(I) is not (s), so m(I) is not (s). For any is (s) if and only if p is a-finite and, index set 1, c0(I) is (r) and (s). M(p) is (s) only if there do not exist infinitely if and only if M(p) is (r). Day (51.
many disjoint sets of positive p-measure, i.e., only if M(p) is finite dimensional. C(K(f)) is (r). Day [61 showed that every (AL)-lattice is (r). We can apply the smoothability theorem given above and these results to conclude the non-
existence of linear maps of a certain kind. There can be no linear mapping I uncountable, with p > 1 into a dense subspace of of any 1,(I) or
or a dense subspace of m0(I), I infinite, or of M(p), if M(p) is not finitedimensional. However, Klee (3] proves that if I is an infinite set, then the is rotundable in such a manner that every Banach space Banach space having a dense subset of cardinality card (I) is congruent to a quotient with this rotund norm. Wada [1] shows that if X is a metric space, then C(X) is (r)
space of
[(s)]
if and
only if X is separable (compact]. VI. Duality. In reflexive spaces certain pairs of geometric properties are in duality, i.e., the conjugate or the space itself has one of the properties if and only if the space itself or the conjugate has the other. In general, no pair of geometric properties are in complete duality unless each is strong enough to imply reflexivity. For this reason, in the absence of reflexivity there are only fragmentary implications from one to the other of the pair. We shall attempt to collect these implications and show how in the reflexive case, complete duality is øbtained. However, we leave to the reader the problem of using the equivalences between properties which were stated in § HI to obtain further pairs of dual properties that are not given here. The theory of dual properties in modulared vector lattices is also sketched in this section. It will be helpful in organizing the results of this section to center these results around the concept of differentiability of the norm. There are four different kinds of differentiability of the norm: (G) Gateaux differentiability,
(UG) uniform Gateaux differentiability, (F) Fréchet differentiability, and (UF) uniform Fréchet differentiability; and each of these in the reflexive case gives rise to a different level of duality.
Smulian (11 showed that if the unit ball in a Banach space B is weakly compact, then B* is (G) if and only if B is (R); and if every continuous linear
functional attains its supremum on the unit ball of B, then B is (G) if and only if B is (R). Thus if B is reflexive, (G) and (R) are in complete duality. From the equivalence between (G) and (S) in any Banach space it follows that in the reflexive case there is complete duality between (S) and (R). Without reflexivity the best that can be said is given by Klee [21: if B* is (R) then B is (S) and if B is (R). According to the results in is not (s). Thus B can be (R) with B* not is (r) but = § V, (S). In [7, p. 114] Day observed that is (R) ((S)1 if and only if every two-
dimensional quotient space of B is (S) ((R)]. This result coupkd with a
D. F. CUDIA
90
proposition of Klee [3] which states that if B is a separable normed Linear space and L is a nonreflexive closed subspace of B, of deficiency 2, then B admits a norm under which B is (S) but BIL is not (S), shows that B can be CS) but
not (R).
Phelps [4] gives the following theorem: A Banach space B is (U) if and only if every weakt closed subspace in Bt is a Chebyshev set. Thus in a reflexive space, B is (U) if and only if Bt is (C) and B is (C) if and only if Bt is (U). Taylor [1] and Foguel [1], between them, show that B is (U) if and only if 8* is (R). These matters are considered further in § VII. Here we merely wish to note the duality between (U) and (C) in reflexive spaces. In [21 Smulian states that B is (UG) if and only if Bt is (WtUR), and that Bt is (UG) if and only if B is (WUR). If B is reflexive then "Bt is (WtUR)" is (WUR)" so that in such spaces (UG) and (WUR) are is equivalent to dual geometric properties.
Smulian [2] proved that Bt is (F) if and only if B is (D) and that B is (F) if and only if Bt is (Dt). In a reflexive space (D) coincides with (D) so that in such a case, there is full duality between (F) and (D). We saw in § Ill that in a Banach space (D) is equivalent to reflexivity of the space and (PLUR);
hence the properties (F) and (PLUR) are dual when the space is reflexive. B is (PLIJR) if and only if B is (H) when B is reflexive. Thus for reflexive spaces there is full duality between (F) and (H), an observation first made by Anderson LI].
Lovaglia [1] showed that if Bt is (LUR) then B is (F), and if B is reflexive then B is (LUR) implies that B' is (F). However to reverse the implications Lovaglia [1] uses an additional hypothesis: if B' is (F) and B is (WLUR) then is (LUR). B is (LUR); if B is reflexive and B is (F) with Bt (WLUR) then Indeed, Anderson [I] exhibits a reflexive space B such that Bt is (F) but B is not (LUR). Anderson [1] showed that if 8* is (F) then B is (MLUR) and if B is reflexive then B is (F) implies that Bt is (MLUR). But (MLUR) appears "too weak" to yield duality with (F) while (LIJR) is "too strong." As shown in § III, in a reflexive space, (H) is "between" (LUR) and (MLUR) and, as it turns out, is dual to (F) in reflexive spaces. For a more systematic development with proofs of the dual geometric theory of norm differentiability see Cudia [I]. Smulian (21 showed that B is (UF) if and only if Bt is (IJR). Since either
property implies reflexivity, (Smuljan [2], Milman (I] or Pettis [1]) there is complete duality between (UF) and (UR). In [41, Day uniformized the relation between smoothess and rotundity of the conjugate and rotundity and smooth• ness of all two dimensional quotient spaces to obtain complete duality between (UR) and (US). No property dual to (kR) is known.
Let (L, m) be a modulared vector lattice and let Lt and L Lt denote the totality of all linear functionals bounded under the norm and all continuous linear functionals xt which are also universally continuous, i.e., for any XA converging monotonicatly downward to 0, we have
ROTUNDITY
inflx*(xA)I =0 A
On
V the associated modular m* of m is defined by the formula m*(x*) = (x(x) — m(x)}
L*. Nakano [1, § 381 shows that m* satisfies all the modular conditions for given in I. The restriction of m* to L is called the conjugate modular of
m and denoted by th. Then (L, ñz) is a modulared vector lattice called the conjugate space of (L, m). In this section we always assume semi-regularity of L, i.e., for any nonzero
x in L there is i in L such that i(x) *
0. By semi-regularity of L the following formulas are valid (see Nakano El, §39—40; 2, §83)),
sup {i(x)
m(x)
—
L
for xeL; lIxII
= sup Ii(x)I
for all XE L. The first norm and the modular norm of by the conjugate modular ,ñ are denoted by II II and III 1111 respectiveLy. Then we have that
for any ieL,
sup
Ii(x)I
and 1111111= sup
Ando [21 proved that m is strictly convex [even) if and only if m is even [strictly convex]. m is said to be uniformly increasing if
X) supinf
lIIxIPI
Nakano (1, § 51] proved if m is uniformly convex Euniformly even) and uniformly increasing then rn is uniformly even [uniformly convex).
These theorems together with those of § III give conditions on the modular sufficient to insure that the first or second norms by the conjugate modular be rotund, smooth, uniformly rotund, or uniformly smooth. As an example note that (Nakano [1, § 51]) if a modular m is uniformly even and uniformly increasing, then the first norm by the conjugate modular th of m is uniformly rotund.
VII. Geometry and reflexivity. Throughout this section B is a Banach space. We wish to examine some relationships between geometry and reflexivity. Dixmier [1) has shown that if B**** is (R) then B is reflexive. Thus if the fifth conjugate is smooth, B is reflexive. Smulian [2] showed that if B is (UF) or if B* and B** are (F) then B is reflexive. Anderson [1] combines a result of Fan and Glicksberg (2, Theorem 3] with a theorem of
92
D. F. CUDIA
Lovaglia [11 that if B* is (LUR) then B is (F) to obtain the result:
B1
is
(LIJR) implies B is reflexive. In this connection Phelps [lj has exhibited a non-reflexive Banach space whose first conjugate is (LUR). That (ur) implies reflexivity, (see Ringrose El) for a short proof), has long been known. Hence B* is (UR) implies that B is reflexive. It would be interesting to know if (ML1JR) of the third conjugate implies reflexivity. Fan and Glicksberg 11) prove that B is (kR) implies B reflexive and show that a (WkR) space is reflexive if and only if it is weakly complete. They also show the existence of (W2R) Banach spaces which are nonreflexive.
Day El] showed that there are reflexive (AB)-lattices which are not (ur). Ando 12] has shown that the situation is quite different with respect to modulared vector lattices. Such a space with its modular norm is reflexive if and only if it is (ur and us). This shows that not every complete (AB)lattice is mocluLarable, i.e., isomorphic to a modular norm by some modular, a result first obtained by Shimogaki [1). The special character of inodulared vector lattices is further indicated by the example of Ando [2] of a complete (AB)-lattice which is (UR) and (US) but which is not modularable. In the case of modulared vector lattices, if the modular is uniformly convex or uni-
formly even, the space is reflexive when normed by either the first or the second norm.
It is well known that for a reflexive Banach space each continuous linear functional attains its supremum on the unit sphere; i.e., each supporting hyperplane of the unit sphere has a point of support. Klee [1] showed that B is reflexive if each continuous linear functional attains its supremum on the unit sphere of any isomorph of the space, a result first obtained by James (2] for spaces with a basis. James 11) proved that a separable Banach space is reflexive if and only if each supporting hyperplane of the unit ball has a point of contact and then he [5) removed the condition that the Banach space be separable. Phelps [1] and Bishop and Phelps [1) between them show that a Banach space with a smooth conjugate is reflexive if and only if every hyper. plane of support of the unit ball has a point of contact. The normed spaces point of view has been found useful in the theory of approximation. For a general survey of the situation see Buck (2]. We shall be concerned only with those aspects having a direct bearing on rotundity and reflexivity. We shall be interested in characterizing the Chebyshev sets in terms of the geometry of the space. Jessen (I] proved that the class of Chebyshev sets coincides with the class of closed convex sets in an n-dimensional Euclidean space. Busemann [1) characterizes rotund finite dimensional Banach spaces as
those in which every closed convex set is a Chebyshev set. The reverse inclusion characterizes finite dimensional Banach spaces which are smooth. Thus the class of Chebyshev sets coincides with the class of closed convex sets
in a finite dimensional Banach space if the space is smooth and rotund. In this connection see the papers of Efimov and
and Konstantinesku El).
As might be expected the infinite dimensional situation is more delicate. In a reflexive space every closed convex subset is an existence set. An
ROTUNDITY
93
argument of James given in Phelps 141 shows that a space is reflexive if and only if every closed subspace is an existence set. Thus a Banach space is
rotund and reflexive if and only if every closed convex set is a Chebyshev set. This extends the first theorem of Busemann cited above to infinite dimensional Banach spaces. In [5] Klee shows that in a smooth reflexive Banach space, a Chebyshev set must be convex if the associated metric pro jection (carrying each point of the space onto that point of the set which is nearest to it) is both continuous and weakly continuous. In the same paper it is shown that if B is (UR) and (UG) then the class of closed convex sets coincides with the class of weakly closed Chebyshev sets. As can be seen much work remains to be done in characterizing the infinite•dimensional Banach spaces in which every Chebyshev set is convex. In a different vein recall that the rotund spaces are those whose unit spheres contain no line segments. Thus the rotund spaces are precisely those for which every one dimensional subspace is a Chebyshev set. Phelps [4] has obtained a similar characterization for smoothness by considering subspaces M having the unique extension property. A short proof in Phelps (4] gives the following theorem: A closed subspace has the unique extension property if and only if its annihilator is a Chebyshev set in the dual. A normed space is smooth if and only if every point on the unit sphere has a unique hyperplane of support. This condition is equivalent to the requirement that every one dimensional subspace have the unique extension property. Applying the theorem stated above we see that a space is smooth if and only if every weak* closed maximal subspace in the dual is a Chebyshev set. In a Banach space with a smooth conjugate, every maximal closed subspace is a Chebyshev set if and only if every closed subspace is a Chebyshev set.
This complements the corresponding statement fot the unique extension prQperty in any Banach space given in Phelps [4]. BIBLIOGRAPHY
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Efirnov, N. V., and Stebkin. S. B. 1. Certain properties of Chebyshev sets, Doki. Akad. Nauk SSSR 118 (1958), 17-19. (Russian); MR 20 $1947. 2. Chebyshev sets in Banach space., Doki. Akad. Nauk SSSR 121 (1958), 582-585 (Russian); MR 20 $6026. 3. Support properties of sets in Banach spaces and Chebyshev sets, Doki. Akad. Nauk SSSR 127 (1959), 254-257. (Russian); MR 21 $5883. Ewald, G. and Kelly, L. M. 1. Tangents in real Bano.ch spaces, J. Reine Angew. Math. 202-203(1959-60). 160-173. Fan, K., and Glicksberg, I. 1. Fully convex normed linear spaces, Proc. Nat. Acad. Sd. U.S. A. 41(1955), 947-953. 2. Some geometric properties of the spheres in a normed linear space, Duke Math. J.
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(1958), 553-568.
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a theorem by A. E. Taylor, Proc. Amer. Math. Soc. 9 (1958), 325.
Fortet, R. 1. Reinarquea sur los espaces uniformétnent convex, (1940), 497-499.
2.
C. R. Acad. Sci. Paris 210
Remarques aur tee espaces uniformé,nent convexes, Bull.
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(1941), 23-46.
Hirschfeld, R. A. 1. Sur La theorie génirale dee meiUeures approxi,nat ions, C. R. Acad. Sci. Paris 246 (1958), 1485-1488.
James, R. C. 1. Reflexivity and the
of linear funetionats, Ann. of Math. (2) 66 (1957),
159-169.
2. Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1960), 518-527. 3. A Banach apace isometric with its second conjugate, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 174-177. 4. Banack spaces with a specified number of separable conjugate spaces, Amer. Math. Soc. Notices 5 (1958), 680. 5. Characterization of reflexivity. Studia Math. (to appear). Jessen, B. 1. Two theorems on convex point sets, Mat. Tidsskr. B. 1940, 66-70. (Danish); MR 2, 261. Kadec, M. 1. 1.
On weak and norm convergence, Doki. Akad. Nauk SSSR 122(1958), 13-16. (Russian);
MR 20 $5422. 2. On spaces which are isomorphic to locally uniformly convex spaces, lzv. Zav. No. 6 (1959), 51-57. (Russian) Kadison, R. V. 1. Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325-338. Kakutani, S. 1. Weak topology and regularity of Banach spaces, Proc. Imp. Acad Japan 15 (1939). 169-173.
KIee, V. L., Jr. 1. Some characterizations of reflexivity, Rev. Ci. (Lima) 52 (1950), 15-23. 2.
Convex bodies and periodic horaeomorphigms in Hubert space, Trans. Amer. Math. Soc. 74 (1953), 10-43.
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Some new results on smoothness and rotundity in normed linear spaces, Math Ann. 139 (1959), 51-63.
4. 5.
Mappings into normed linear space,, Fund. Math. 49 (1960), 25-34. Convexity of Chebyshev sets, Math. Ann. 142 (1961), 292-304. Konstantinesku, F. 1.
On Chebyshev sets, Doki. Akad. Nauk SSSR 29 (1960), 21—22. (Russian); translated as Soviet Math. DokI. 1 (1960), 11-12. Kra&ovskii, S. N. and Vinogradov, A. A. 1.
On a criterion of uniform convexity of a space of type B, Uspehi Mat. Nauk
(N. S.) 7 (1952), no. 3 (49), 131-134. (Russian); MR 14, 55. Laugwitz. D.
Grundlagen für die Geometrie der unendiichdimenstonaien Finsier-Räu,ne, Ann. Mat. Pura AppI. 41 (1956), 21-41. Lorch, E. R. 1. A curvature etudy of convex bodies in Banach spaces, Ann. Mat. Pura AppI. (4) 1.
34 (1953), 105-112.
Lovaglia, A. R. 1.
Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78(1955), 225-238.
Luxemburg, W. A. J. 1. Banach function spaces, Doctoral Dissertation, University of Delft, 1955. McShane, E. J. 1. Linear functionals on certain Banach spaces, Proc. Amer. Math. Soc. 1 (1950), 402-408.
Mazur, S. 1.
tiber konvexe Mengen in tinearen normierten Rdusnen, Studia Math. 4(1933), 70-84.
2. (Jber schwache Konvergenz in den Räumen (L'), Studia Math. 4 (1933), 128-133. Milman, D. P. 1.
On some criteria for the regularity of spaces of the type (B), Doki. Akad, Nauk SSSR (N. S.) 20 (1938), 243-246.
Milnes, M. W. 1. Convexity of Orlicz spaces, Pacific J. Math. 7 (1957), 1451-1483. Minkowski, H. 1. Theorie der konvexen Körper, insbesondere Begründung urea Oberflachenbegrjffs, Gesammelte Abhandlnngen, Vol. II, pp. 131-229, Teubner, Berlin, 1911. Nakano, H. 1. Modutared semi-ordered linear spaces, Tokyo Math. Book Series, Vol. 1, Maruzen, 1950.
2. Topology and linear topological spaces, Tokyo Math. Book Series, Vol. 3. Maruzen, 1952.
Nordlander, G. 1. The modulus of convexity in normed linear spaces, Ark. Mat. 4 (1960), 15-17. A. 1.
On the iso,norphism of the spaces m and M, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 0 (1958), 695-696.
Pettis, B. J. 1. A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), 249-253.
Phelps, R. R. 1. Subreflexive norr,sed linear spaces, Arch. Math. 8 (1957), 444-450.
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2. Convex sets and nearest points. U, Proc. Amer. Math. Soc. 9 (1958), 867-873. 3. Some subreflezive Banach spaces, Arch. Math. 10 (1959), 162-169. 4. Uniqueness of Hahn-Banach extensions and unique beet approccimation, Trans. Math. Soc. 95 (1960), 238-255. Riesz, F. and Sz..Nagy, B. 1. Lecons d'awalyse fonctionnelle, Akadémiai Kiadó, Budapest, 1952. Ringrose, J. R. 1. A note on uniformly convex spaces, J. London Math. Soc. 3.4 (1959), 92. Rund, H. 1. The differential geometry of Finsler spaces, Springer, Berlin, 1959. Ruston, A. F. 1. A note on convexity in Banacl& spaces, Proc. Cambridge Philos. Soc. 45 (1949), 157 -159.
Segal, 1. E. 1. Quantization of nonlinear systems, J. of Mathematical Phys. 1 (1960), 468-488. Shimogaki, T. 1. On the norms by uniformly finite ,nodulars, Proc. Japan Acad. 33(1957), 304-309. Smulian, V. L.
some geometrical properties of the unit sphere in the space of the type (B), Mat. Sb. (N. S.) 6 (46) (1939), 77-94. (Russian. English summary); MR 1, 242. 2. Sat La dérivabilité de La norme dane L'espace de Banach, Dokl. Akad. Nauk SSSR 1. On
(N. S) 27 (1940), 643-648. 3. Sat quelques propriétés géometriques de La sphere dana los espaces linéaires se ordonnis de Bausch, DokI. Akad. Nauk SSSR (N. S) 30 (1941), 394-398. 4. Sur La structure de La sphere unitaire dane l'espace do Banach, Mat. Sb. (N.S.)
9 (51) (1941), 545-561. Sw' lea topologies différentes dana L'eapace de Banach, Dokl. Akad. Nauk SSSR (N.S.) 23 (1939), 331-334. Sundaresan, K. 1. Index of flattening, Detailed Reports of Abstracts of the Indian Science Congress, 5.
Bombay, 1960.
Taylor, A. E. 1. The extension of linear functionals, Duke Math. J. 5 (1939), 538-547. Wada, J. 1. Strict convexity and smoothness of normed spaces, Osaka Math. J. 10
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Yamamuro, S. 1. On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90(1959), 291 -311. 2. Exponents of modulated semi-ordered spaces, J. Fac. Sci. Hokkaido Univ. Ser. 1
12
(1953), 211-253.
UNIVERSITY OP ILLINOIS
OF THE CIRCLE1
A BY
LUDWiG W. DANZER
V. Mizel has asked whether the following is true: Assume is a planar closed convex curve and no rectangle has then 1i is a circle. exactly three of its vertices on A first proof of this theorem was given by A. S. Besicovitch [1] and presented by him at the symposium on convexity in Seattle in June 1961. Since his proof employed rather deep results on linear measures, he invited his audience to look for a more elementary proof. I am glad to accept this invitation.
If c is any chord of and one of the lines (both the lines) perpendicular to c and supporting c also supports then we shall call c a normal (doublenormal) of (L Two points of which are joined by a double-normal will be called anlipodal The first to show is: (1) Every normal AB of is a double-normal. but n does not. This Suppose m, n I AB, A e m, B€ n and m supports and different from B (I is would imply the existence of a point C€ n closed). The point D which makes ABCD a rectangle hence would be in m fl I. Since m is a line of support this would mean DA c I. Now by the line n would be found to considering the midpoints of DA and of have three distinct points in common with I. Hence n supports I (I is convex). Contradiction! As a consequence of (1) we have (2) 1 is a curve of constant width, say of width d. Following from (1) and (2) we get: (3) Every normal of I has length d. Though it has been well known probably for decades that (1) implies (2), there seems to be no older a reference than the paper by Besicovitch men-
I
tioned above.
If A1 e I and two different points X and Y both are antipodal to A1, then the circular arc XY (center A1) lies completely on I and its midpoint A2 is on a unique supporting line, whence A1 is the only point antipodal to A2. Thus we have: (4) There exists a point A on I whose antipodal point A* is unique. From now on we shall keep this point A fixed. Our main purpose is to show, (5) If ABCD is a rectangle with B, C, Del then C = A*. Consider the set Research supported by the National Science Foundation, U.S.A.. under Grant NSFG 18975. 99
L. W. DANZER
100
{A')
and 3B: (Be I and AB .L BC)); it is closed and it is easily seen, that A has a positive distance from and B is different from C. Hence there exists a rectangle for every CE \ =
u {C I Ce
all its vertices on I and such that the open arc AB1C1 of I has no point in common with %. Similarly there is a rectangle n such that = 0. This means c C1C, and hence (5) is equivalent ABICIDI with
= containing PRooF (6). Suppose C1 * A. Consider the closed halfplane B1 and being bound by AC1, and also the semicircle with endpoints A and C1 which contains B1. Then n = {C1) and C1 =
In
cony (s). To see the latter, assume the existence of a point B on I n but outside Draw the line p through B perpendicu'ar to AB. Because of (4) (and As e i)), c
p would not support I. Hence there would be a unique point C p
I
different from B and therefore in From this would follow C in contradiction to the C1 e mt (4ABC) (recall that B was assumed outside convexity of I. at B1 supports Now consider B1. Since it is on the line t tangent to (5,. I is perpendicular to coincide, and of ABICIDI Since the centers of Thus B,D1 and this turns out to be a normal of I. (see(3)) rendering
and
(see(4)). C1=A5 Thus (5) is verified. Similarly one proves A5 But, by the main assumption of our theorem, every point of I may serve forB in (5), and this means (Thales): I coincides with the circle of which AA5 is a diameter. It would be interesting to learn whether this theorem remains true when also nonconvex curves are allowed in the competition. Another question is, how to generalize this theorem in a proper way into higher dimensions. REFERENCE
1. A. S. Besicovitch, A problem on a circle, 5. London Math. Soc. 36(1961), 241-244. UNIVERSITY OF WASHINGTON
HELLY'S THEOREM AND uS BY
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
Prologue: Eduard Helly. Eduard Heily was born in Vienna on June 1, 1884. He studied at the University of Vienna under W. Wirtinger and was awarded the Ph. D. degree in 1907. His next few years included further research and study in Göttingen, teaching in a Gymnasium, and publication of four volumes of solutions to problems in textbooks on geometry and arithmetic. His research papers were few in number but contained several important results. The first paper [1] treatçd some basic topics in functional analysis. Its "selection
many applications and is often referred to simply as "Helly's theorem" (see Widder's book on the Laplace transform). The same paper contains also the Helly-Bray theorem on sequences of functions (see principle"
Widder's book) and a result on extension of linear functionals which is mentioned in the books of Banach and Riesz-Nagy. His famous theorem on the intersection of convex sets (also commonly called "Helly's theorem") was discovered by him in 1913 and communicated to Radon. Belly joined the Austrian army in 1914, was wounded by the Russians, and
taken as a prisoner to Siberia, where one of his "colleagues" was T. Rado. He returned to Vienna in 1920, and in 1921 was married and appointed Privatdozent at the University of Vienna. Along with his mathematical research and work at the University, he held important positions in the actuarial field and as consultant to various financial institutions. His paper [3] on systems of linear equations in infinitely many variables was a muchquoted study of the subject. The "Belly's theorem" which is of special
interest to us here was first published by him in
1923 [41
(after earlier
publication by Radon and König), and the extension to more general sets in 1930 [5).
In 1938 the Hellys emigrated with their seven-year-old son to America, where Professor Helly was on the staff of Paterson Junior College, Monmouth Junior College (both in New Jersey), and the Illinois Institute of Technology. He died in Chicago in 1943. For much of the above information we are indebted to his wife Elizabeth (also a mathematician, now Mrs. B. M. Weiss),
who resides at present in New York City. Their son Walter (Ph. D. from M. I. T.) is a physicist with the Bell Telephone Laboratories in New York. 1.
S.-B. Akad. Wiss. Wien 121 (1912),
Uber lineare
265—297.
This work was supported in part by a grant from the National Science Foundation, U. S. A. (NSF-G18975). The authors wish to express their appreciation to the National I
Science Foundation, and also to Lynn Chambard for her expert typing. 101
102
2.
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
(Jber Reilwnentwicklungen nach Funklionen eines Orthogonalsyslems, S.-B.
Akad. Wiss. Wien 121 (1912), 1539-1549. 3.
Uber Systeine linearer Gleichungen mit unendlich vielen Unbekannten,
Monatsh. Math., 31 (1921), 60-97. 4. Uher Mengen konvexer KOrper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Vereiti. 32 (1923), 175-176. 5. Uber Systeme von abgeschlossenen Mengen mit gemeinschafllichen Punkten, Monatsh. Math. 37 (1930), 281—302. 6. Die nene englische Sterblichkeitsmessung an Versicherten, Assekuranz Jahrbuch, 1934.
Introduction. A subset C of a (real) linear space is called convex if and only if it contains, with each pair x and y of its points, the entire line segment [x, yj joining them. The equivalent algebraic condition is that ax + (1 — a)y e C
whenever x E C, ye C, and a e [0, 1]. At once from the definition comes the most basic and obvious intersection property of convex sets: the intersection of anv famil.v of convex sets is again a convex set, though of course the inter-
section may be empty. The present exposition centers around a theorem setting forth conditions under which the intersection of a family of convex sets cannot be empty. This famous theorem of Eduard Helly may be formulated as follows: THEOREM. Sufrpose is a family of at least n + 1 convex sets in affine
R", and
if each n + 1 members of to all members of .51'.
is
finite or each member of .2 is compact. Then have a common point, there is a Point common
FIGURE
1
Let us inspect two simple examples. Consider first a finite family of compact
convex sets in the line R', each two of which have a common point. Each set is a bounded closed interval. If [a1 , are the sets of the . . .,
THEOREM AND ITS RELAT1VES
family, it is clear that the point mm1 {,9,) is common to all of them, as is the point max1 (a1) and of course all points between these two. Consider next a family of three convex sets in R2, having a common point (see Figure 1). This gives rise to three shaded areas which are the pairwise intersections of
the sets and to a supershaded area which is the intersection of all three. Helly's theorem shows that if a convex set in R2 intersects each of the three shaded areas, then it must intersect the supershaded area. The convex hull cony X of a set X in a linear space is the intersection of all convex sets containing X. Equivalently, cony X is the set of all convex combinaf ions of the points of X. Thus p e cony X if and only if there are (t'1 1 points x1, •, x, of X and positive numbers a:, •, a,,, such that ax,,. Helly's theorem is closely related to the following results and p = on convex hulls: When X c each point of COUV X j5 a CARATHEODORY'S THEOREM. combination of n + 1 (or fewer) points of X. RADON'S THEOREM. Each set of n + 2 or snore Poi,zts in can be expressed point. as the union of two disjoint sets whose convex hulls have a
Consider, for example, a set X c R2. By Carathéodory's theorem, each point
of cony X must be a point of X, an inner point of a segment joining two points of X, or an interior point of a triangle whose vertices are points of X. If X consists of four points, Radon's theorem implies that one of the points lies in the triangle determined by the other three, or the segment determined by some pair of the points intersects that determined by the remaining pair (see Figure 2).
FIGURE
2
The theorem of Carathéodory was published in 1907. Helly's theorem was discovered by him in 1913, but first published by Radon [21 in. 1921 (using Radon's Theorem). A second proof was published by Konig in 1922, and Helly's own proof appeared in 1923 (Helly [1]). Since that time, the three
104
LUDWIG DANZFR,
GRUNBAUM, AND VICTOR KLEF
theorems, and particularly that of Helly, have been studied, applied, and generalized by many authors; especially in the past. decade has there been a
steady flow of publications concerning Helly's theorem and its relatives. Many
of the results in the field (though not always their proofs) would be understandable to Euclid, and most of the proofs are elementary, as is true in most parts of combinatorial analysis which have not been extensively formalized. Some of the results have significant applications in other parts of mathematics,
and there are many interesting unsolved problems which appear to be near the surface and perhaps even accessible to the "amateur" mathematician. Thus
the study of Helly's theorem and its relatives has several nontechnical aspects in common with elementary number theory, and provides an excellent introduction to the theory of convexity. The present report is intended to be at once introductory and encyclopedic. Its principal aim is to supply a summary of known results and a guide to the literature. Contents are indicated by section headings, as follows: Proofs of Helly's theorem; Applications of 1-lelly's theorem; The theorems of Carathéodory and Radon; Generalizations of Helly's theorem; Common transversals; Some covering problems; Intersection theorems for special families; Other intersection theorems; 9. Generalized convexity. Since the report is itself a summary, it seems pointless here to summarize the contents of the individual sections. The emphasis throughout is on comFor interbinatorial methods and hence on finite families of subsets of section properties of infinite families of noncompact convex sets, especially in infinite-dimensional linear spaces, see the report by Klee 16). 1-2 most results are Many unsolved problems are stated, and after stated without proof. The bibliography of about three hundred items contains all papers known to us which deal with Helly's theorem or its relatives in a finite-dimensional setting. In addition, we list many other papers concerning intersection or covering properties of convex sets, and some general references for the study of convexity and generalized convexities. (We are indebted to E. Spanier and R. Richardson for some relevant references in algebraic topology, and to J. Isbell for information about the paper of Lekkerkerker-Boland 111.) Some of the material treated here appears also in books by BonnesenFenchel [II, Yaglom-Boltyanskji [1]. Hadwiger-Debrunner 131, Eggleston [3), Karlin (1), and in the notes of Valentine [8j. In general, these will be referred to only for their original contributions. For fuller discussion of some of the unsolved problems mentioned here (and for other elementary problems), see the forthcoming book by Hadwiger-Erdös-Fejes Tóth-Klee [1). Organization of the paper is such that formal designation of various results as lemma, theorem, etc., did not seem appropriate. However, the most 1.
2. 3. 4. 5. 6. 7. 8.
important results are numbered, both to indicate their importance and for
HELLY'S THEOREM AND ITS RELATIVES
purposes of cross-reference. In order to avoid a cumbersome numbering system, we adhere to a convention whereby 4.6 (for example) refers to the numbered result 4.6 itself, 4.6 if. refers to 4.6 together with immediately subsequent material, 4.6+ refers to material which follows 4.6 but precedes 4.7, and 4.6— refers to material which precedes 4.6 but follows 4.5. Much of our notation and terminology is commonly used, and should be clear from context. In addition, there is an index to important notions and notations at the end of the paper. Equality by definition is indicated by or =:. When used in a definition, "provided" means "if and only if"; the latter is also expressed by "iff". The set of all points for which a statement P(x) is true is usually denoted by Ix: P(x)). However, when P(x) is the conjunction of two statements P'(x) and P"(x), and P'(x) is of especially simple form beginning "x we sometimes write IP'(x): P"(x)} for IX: P(x)} in the 1): = (x: xE E interest of more natural reading. (For example, {x E: II x and lixil 1).) Though much of the material is set in an n-dimensional real linear space the full structure of RW is not always needed. Some of the results are available for finite-dimensional vector spaces over an arbitrary ordered field, while others seem to require that the field be complete or archimedean. We have not pursued this matter. The n-dimensional Euclidean space (with its usual metric) is denoted by is a compact convex set with nonempty interior. It is A convex body in smooth provided it admits a unique supporting hyperplane at each boundary point, and strictly convex provided its interior contains each open segment Jx,v[ joining two points of the body. The family of all convex bodies in is denoted by A flat is a translate of a linear subspace. The group of all translations in is denoted by T", or simply by T when there is no danger of confusion. A positive homothety is a transformation which, for some fixed ye and some real a > 0, sends x R* into y + ax. The group of all positive homotheties 1
in
is denoted by H" (or simply H) and the image of a set X under a
positive homothety is called a homothet of X. Set-theoretic intersection, union, and difference are denoted by fl U, and respectively, + and — being reserved for vector or numerical sums and differences. The intersection of all sets in a family is denoted by For a point xe R", a real number a, and sets X and Y c R", aX: = (ax: x E X}, x Y: = {x+y :v€ YJ, X+ Y: (x fy : XE X,ye YJ, TX is the family of alt
translates of X, and HX is the family of all homothets of X. The interior, closure, convex hull, cardinality, dimension, and diameter of a set X are denoted respectively by mt x, ci X, cony X, card X, dim X, and diam X. The
symbol 0 is used for the empty set, 0 for the real number zero as well as for the origin of R". The unit sphere of a normed linear space is the set {x: lIxIl = 1), while its unit cell is the set (x: IIxII 1). The n-dimensional Euclidean unit cell is denoted by B", the unit sphere (in by S". A cell in a metric space (M, p) is a set of the form {x: p(z, x) c) for some z€ M and > 0. When we are working with a family HC for a given convex
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
106
body C
in
these
homothets are also called cells. When p denotes a distance
function and X and Y are sets in the corresponding metric space, p(X, Y): = inf{p(x,y):xe Y,y€ Y}. In general, points of the space are denoted by small Latin letters, sets by capital Latin letters, families of sets by capital script letters, and properties of families of sets (i.e., families of families of sets) by capital Gothic letters. Small Greek letters are used for real numbers, indices, and cardinalities, and sometimes small Latin letters are also used for these purposes. Variations from this notational scheme are clearly indicated. 1.
Proofs of Helly'8 theorem. As stated in the Introduction, Helly's theorem
deals with two types of families: those which are finite and those whose members are all compact. Though the assumptions can be weakened, some care is necessary to exclude such families as the set of all intervals ]O, j9] for > 0 or the set of all half-lines [a, oo[ c R'. For a family of compact convex sets, the theorem can be reduced at once to the case of finite families, for if the result be known for finite families then in the general case we are faced with a family of compact sets having the finite intersection property (that is, each finite subfamily has nonempty intersection), and of course the intersection nf such a family is nonempty. This is typical of the manner in which many
of the results to be considered here can be reduced to the case of finite families. The essential difficulties are combinatorial in nature rather than topological, and whenever it seems convenient we shall restrict our attention to finite families. The reader himself may wish to formulate and prove the extensions to infinite families. (See also Klee [6] for intersection properties of infinite families of noncompact convex sets.)
For a finite family of convex sets, Helly's theorem may be reduced as follows to the case of a finite family of (compact) convex polyhedra: Suppose is
a finite family of convex sets (in some linear space), each n + 1 of
which have a common point. Consider all possible ways of choosing n + 1 members of ,%", and for each such choice select a single point in the intersection of the n + 1 sets chosen. Let I be the (finite) set of all points so
selected, and for each KG let K' be the convex hull of K J. It is evident that each set K' is a convex polyhedron, that each n + 1 of the sets K' have a common point, and that any point common to all the sets K' must lie in the intersection of the original family .2'•. We turn now to some of the many proofs of Helly's theorem, with apologies to anyone whose favorite is omitted. (Some other proofs are discussed in 4 and 9.) It seems worthwhile to consider several different approaches to the theorem, for each adds further illumination and in many cases different approaches lead to different generalizations. Helly's own proof (1] depends on the separation theorem for convex sets and proceeds by induction on the dimension of the space. (Essentially the same proof was given by Konig (Il.) Among the many proofs, this one appeals to us as being most geometric and intuitive. The theorem is obvious ri
for R°. Suppose it is known for R" ', and consider in
a finite family
of
HELLY'S THEOREM AND ITS RELATIVES
107
at least n + 1 compact convex sets, each n + 1 of which have a common point. of is empty. Then there are a subfamily Suppose the intersection
.5t' and a member A of9 such thatir..9 = 0 but ,r(fr 0. Since A and M are disjoint nonempty compact convex subsets of the such separation theorem guarantees the existence of a hyperplane H in that A lies in one of the open halfspaces determined by H and M lies in the other. (To produce H, let ii II be a Euclidean norm for let x and y be pointsof A and Mrespectively such that lix —yli = p(A,M): = inf{lia — mu:
A, me M}, and then let H be the hyperplane through the midpoint (x + y)/2 which is orthogonal to the segment [x, y]. (See Figure 3.) Now let J denote a
the intersection of some n members of {A). Obviously J M, and since each n + 1 members of have a common point, I must intersect A. .
FIGURE
3
Since / is convex, in extending across H from M to A it must intersect H, and thus there is a common point for each n sets of the form G fl H with C e j7 — {A). From the inductive hypothesis as applied to the (n — 1)dimensional space H it follows that M fl H is nonempty, a contradiction completing the proof. Radon's proof [2] is based on the result stated above as Radon's theorem. To prove this result, suppose p1. . ., p,,, are points of R" with m n + 2. Consider the system of n + 1 homogeneous linear equations, .
=0= where
=
in the usual coördinatizatjon of
(1
n),
Since m > n + 1,
108
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
the system has a nontrivial solution (r1, •, r,,,). Let U be the set of all i > 0. for which r, 0, V the set of all i for which r, < 0, and c: = completing the proof = LEE Then Lev = —c and of Radon's theorem. we Now to prove Helly's theorem for a finite family of convex sets in observe first that the theorem is obvious for a family of n + 1 sets. Suppose with j the theorem is known for all families of j — 1 convex sets in sets, each n + 1 of which convex a family of j + 2, and consider in n there have a common point. By the inductive hypothesis, for each A — {A}, and by Radon's theorem is a point p.4 common to all members of such that some point and into subfamilies there is a partition of (see Figure z is common to the convex hulisof {p,:FejT} and and the proof is = {B, D)). But then z e = {A, C) and 4, where .
complete.
FIGURE 4
The literature contains many other approaches to Helly's theorem. Some of the more interesting may be described briefly as follows: Rademacher-Schoenberg ni and Eggleston (3] employ Carathéodory's theorem and a notion of metric approximation. Sandgren [11 and Valentine [91 employ Carathéodory's theorem and the duality theory of convex cones; Lannér's use of support functionals [1] leads also to application of the duality theory. Bohnenblust-Karlin-Shapley (1] employ Carathéodory's theorem to prove a result on convex functions from which Helly's theorem follows (see 4.5).
HELLY'S THEOREM AND ITS RELATIVES
Hadwiger [2; 5) obtains Belly's theorem and other results by an application
of the Euler-Poincaré characteristic, which he develops in an elementary setting.
Belly [2] proves a topological generalization by means of combinatorial
topology (see 4.11).
R. Rado [21 proves an intersection theorem in a general algebraic setting and deduces Helly's theorem as a corollary (see 9.4). Levi's axiomatic approach [11 is based on Radon's theorem (see 9.3). Additional proofs of Helly's theorem are by Dukor [11, Krasnosselsky [21, Proskuryakov [1), and Rabin [1). For our taste, the most direct and simple approaches to Belly's theorem are those of Helly and Radon described earlier.
However, each of the many proofs throws some light on the theorem and related matters which is not shed by others. As is shown by Sandgren ['11 and Valentine [9), the duality theory provides efficient machinery for study of Helly's theorem and its relatives. The approach by means of combinatorial topology leads to many interesting problems but remains to be fully exploited. We have indicated the close relationship of Helly's theorem to the theorems of Carathéodory and Radon. In fact, each of these three theorems can be derived from each of the others, sometimes with the aid of supporting hyperplanes and sometimes without this aid, and each can be proved "directly" by means of induction on the dimension of the space (with the aid of the support or separation theorem). Perhaps the inter-r&ationships could best be understood by formulating various axiomatic settings for the theory of convexity,
and then studying in each the interdependence of these five fundamental results: Helly's theorem, Carathéodory's theorem, Radon's theorem, existence
of supporting hyperplanes at certain points of convex sets, existence of separating hyperplanes for certain pairs of convex sets. Levi [1] makes a small step in this direction, and § 9 describes several generalized convexities from which the problem might be approached. This seems a good place to state a sort of converse of Belly's theorem, due provided to Dvoretzky [1). Let us say that a family of sets has the the intersection of the entire family is nonempty or there are n + 1 or fewer sets in the family which have empty intersection. Helly's theorem asserts that each family of compact convex sets in R' has the Of course the does not characterize convexity, for a family of nonconvex sets in R* may have the "by accident" (see Figure 5). However, Dvoretzky's theorem may be regarded as saying that if a family of compact sets in R" has the ON-property by virtue of the linear structure of its members, then all the members are convex. DVORETZKY'S THEOREM. Suppose {K,: e I} is a family of compact sets in none of which lies in a hyperplane. Then the following two assertions are
equivalent:
all the sets K, are convex;
(for each c) J, is affinely equivalent to K,, then the family {J: c 6 fl has the
110
LUDWIG DANZER, BRANKO GRIJNBAUM, AND VICTOR KLEE
FIGURE
5
An example may be helpful. Each of the families 5a, 5b, and Sc in Figure 5 consists of four sets (all but one convex), obtainable by suitable translations because from the members of 5a. The families Sa and 5b have the of their position rather than by virtue of the linear structure of their members. The family Sc lacks the 2. Appllcation8 of Helly's theorem. We may distinguish two types of applications of Helly's theorem, although the distinction is somewhat artificial. It is used to prove other combinatorial statements of the general form: if a certain type of collection is such that each of its k-membcred subfamilies has a certain proPerly, then the entire collection has the property. And it is used to
prove theorems which are not explicitly combinatorial in statement, but in which a property of a class of sets is established by proving it first for certain especially simple members of the class, and then stepping from this special case to the general result by means of Helly's theorem. The same description applies to applications of Carathéodory's theorem. We shall give several examples of applications of Helly's theorem, with references and comments collected at the end of the section. 2.1. Suppose is a family of at least n + I convex sets in R", C is a convex set in R", and .2' is finite or C and of . are Then the existence of some translate of C whiclz intersects [is contained in;
containsl all members of . for each n + I members of
is
guaranteed by the existence of such a translate
PRoOF. For each Ke let K' :(x + C)rK), where r means "intersects" or "is contained in" or "contains". Then each set K' is convex and the above hypotheses imply that each n + I of the sets K' have a common point. By Helly's theorem, there exists a point ze K' and then (z ± C)rK for all Ke A common transversal for a family of sets is a line which intersects every set in the family.
2.2.
Let .9 be a finite family of parallel line segments in R2, each three of
HELLYS THEOREM AND ITS RELATIVES
111
which admit a common transversal. Then there is a transversal common to all members of .5?'. PROOF. We may suppose ..9' to consist of at least three members and that all segments are parallel to the V-axis; for each segment Se .5?' let Ca denote
the set of all points (a, Then each set
ax +
C....
R such that S is intersected by the line y — is cdnvex and each three of such sets have a
common point, whence by Helly's theorem there exists a point (a0 , fto) C fl The line y = a0x + is a transversal common to all members of .5/'. 2.3. If a convex set in R2 is covered by a finite family of open or closed halfspaces, then it is covered by some n + 1 or fewer of these halfspaces. PROOF. Suppose, more generally, that ,sr is a finite family of sets in covering a convex set C, and that for each F€ .5 the set F': = C F is convex. Then (F': Fe is a finite family of convex sets whose inter-
section is empty, so by Helly's theorem there are n + 1 or fewer sets in this family whose intersection is empty. This completes the proof. 2.4. Two finite subsets X and Y of R2 can be strictly separated (by some hyperplane) if and only if for every set S consisting of at ,nost n + 2 points from X U V. the sets S fl X and S fl V can be strictly separated. PROOF. We may assume that X U Y includes at least n + 2 points. For each x: = (x', ..., x") X and y: = (y', , y%) e V. let the open halfspaces f and Q, in R"' be defined as follows: .
/2: =
{'A
= {A: By
= (A,, A,, . .., +
+
:
> o}
tA,y' < o}.
hypothesis, each a + 2 members of the family {J2:xcX} U {Q,:ye Y)
have a common point, and hence by Helly's theorem there exists A e Q,).
Then
fl
the sets X and Y are strictly separated by the hyperplane
(a €R2:
= When u and v are points of a set X c R2, v is said to be visible from u (in
X) provided Eu, vi C X. 2.5. Let X be an infinite comPact subset of and suppose that for each a + I Points of X there is a Point from which all n + 1 are visible. Then the set X is sfarshaped (that is, there is a Point of X from which all Points of X
are visible). PROOF. For each x€ X, let V2 = {y: [x,v) c The hypothesis is that each n + 1 of the sets V2 have a common point, and we wish to prove that V2 * 0. By Helly's theorem, there exists a point ye cony V2, and we shall prove that ye V2. Suppose the contrary, whence there exist XE Xand u e[y, x[ X, and there exists x' X fl [U, with (u,x'[ A X = 0.
Further, there exist w e
x'[ such that II
w
—
x'
= (1/2)p({u}, X), and v
Eu, wJ
LUDWiG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE
112
and x0eX such that fix0 —
= p([u, w], X). Since x0 is a point of Xnearest lies in the closed halfspace Q which misses v and
vu
to v, it is evident that is bounded by the hyperplane through x0 perpendicular to fv, xo]. But then Since p({v}, X) y e cony c Q and Lyx0 ir/2, whence Lx0vy < X) < p({u}, X), it is clear that v * u and hence some point of Lu, z4 is p({w}, closer to x0 than v is. This contradicts the choice of v and completes the proof.
The preceding five results have all illustrated the first type of application of Helly's theorem. We now give three applications of the other sort, in which the theorem is not apparently of combinatorial nature, but nevertheless Helly's theorem is very useful. 2.6. If X is a set in with diam X 2, then X lies in a (Euclidean) cell If X does not lie in any smaller cell, then ci X conof radius [2n/(n ÷ tains the vertices of a regular n-simplex of edge-length 2. PROOF. We present two proofs of this important result. The first is logically simpler and shows how the theorems of Helly and Carathéodory can sometimes substitute for each other in applications. The second is more under-
standable from a geometric viewpoint. By combining aspects of these two proofs, one can arrive at a third which is easily refined to yield the second equality of 6.8 below.
By Heily's theorem (or 2.1), 2.6 can be reduced to the case of sets of n + 1, and for each cardinality + 1. For consider Xc E" with' card = {y: fly — xfl < [2n/(n + If 2.6 is known for sets XEX the cell n+1 the sets have a common point, of cardinality is nonempty by Helly's theorem and the desired conclusion whence follows.
n + 1. Let y denote the center of the
Now suppose X c E" with card X
smallest Euclidean cell B containing X and let r(X) be its radius. With = r(X), let
where m n. It is easily verified that y cony (a0, .. ., a,,) and we assume with without loss of generality that y = 0, whence 0 = 1 and — 2, whence = 2r2 let di,: 0. For each i and always 2(z., z,). For each j, 1
—
a= = r212
and
zi)/2 = r212,
—
summing on j (from 0 to m
(m + 1)r/2, whence
r
n) leads to the conclusion that m
—
2m
m+1
2n
n+1
HE[.LY'S THEOREM AND ITS RELATIVES
Further, equality here implies that m =
= 2 for all i * j, so the
n and
proof is complete. In the above paragraph, the assumption card X n + 1 (justified by Helly's theorem) was used only to insure that the point ye cony X could be expressed
as a convex combination of n + 1 or fewer points of X. But this is also insured by Carathêodory's theorem, so the above proof could also be based on the latter. The second proof of 2.6 is by induction on the dimension. For E', the theorem is trivial. Suppose it is known for E"1 and consider a set Xc with card 1. Let y,B, and r(X) be asabove. IfyeconvZforsome Z X, then the proof is completed by the inductive hypothesis, for B is then the smallest cell containing Z. (It is easily proved that ify cony Z and pe E' —.(y), there exists z€Z with ilz—yll < lIz —P11. This is equivalent tothe lemma mentioned at the end of 9.9 below.)
Hence we may assume that card X= n + 1 andyeintconvX. Now let. denote the family of all (n + 1)-pointed sets Xc E* (with diam X 2) for which r(X) is maximal. By compactness, 2 is nonempty, and clearly Xe 2' implies diam X = 2. Consider a set {x0, - . ., x,,} .2' and suppose min{Iix. — x11I :0 j <j n} = Ilxo —x111 1, and ---, Ii
the contradiction completes
the proof. 2.7. If C is a convex body in there exists a point z cC such that for each chord Eu, v] of C which passes through z, liz — ulI/iiv — ull n/(n + 1). PROOF.
For each point xc C, let
= x + n(n + 1)1(C — x). We claim that
0, and to prove this it suffices (in view of Helly's theorem) to show that if x0, . ., x,, are points of C, then fl C1, includes the point y: = (n + 11' E x,. This is evident, since for each j it is true that (C—x1). n+1 n n +1 Now consider an arbitrary chord Eu, vi passing through the point z e C1. Then z cu + n(n + ([u, v] — ii), whence z = u + n(n + 1)"2(v — u) for some A 10,11 and liz — uII/lIv — uli = An(n + 1)_1 n(n + 1)-', completing the proof. 2.8. Let be an additive family of sets which includes all oPen halfspaces in and suppose p is a function on JV to [0, oo [ which satisfies the following conditions: a°. If X, Ye then p(X U Y) p(X) + p(Y); b°. c°.
if X, Ye.'/ with Xc Y, then p(X) there is a bounded set Bc
such that
(n + IY'p(R1). Then there exists a Point x* for each open halfspace J containing
and p(R1.—B) < such that
(n + 1)1p(R")
114
LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE
de1. Let PROOF. We assume without loss of generality that let J' denote and for each Je note the set of all open halfspaces in such that denote the set of all G e the closed halfspace I?" J. Let we see from of Then for any n + 1 members G0, p(G) < (n + p(G1) < 1, whence U and n: * 0. (Similarly, a° that p(U has fewer than n -f- 1 members.) It follows from the theorem is trivial if has nonempty Helly's theorem that every finite subfamily of {G': Ge
intersection. Now with B as in c°, it is easy to produce a finite subfamily F' is bounded and contains B. such that the intersection of we have F c R" — F' c — B, whence from b° and For each Fe Thus c° it follows that /L(F) p(R" B) 1. That is, cony X = H(X), where H(X): = It is natural to ask how many times the operation H, must be iterated to produce the convex hull of a set in 1?". The case .1 n + I is settled by Carathêodory's theorem. The case j = 2 was treated by Brunn [11 and in subsequent years by several other authors (Hjelmslev (1), Straszewicz [1], Bonnesen-Fenchel [1], Aix -Kubota-Yoneguchi [I]), However, the question
becomes trivial (modulo Carathêodory's theorem) in view of the following
almost obvious fact: 3.4.
= H,k(X).
HELLY'S THEOREM AND ITS RELATIVES
From this it follows (as noted by Bonnice-Klee [1]) that if X C R" and J&fz = c.onv X; conversely, if X is the set of all n ÷ 1, then H,1(H,2 (H,,1(X)) •) * cony X. vertices of an n-simplex and f112 j1 n, then H,L(Hjz (Related problems on the generation of affine hulls are much more complicated. See Klee [8].)
Carathéodory's theorem is the best possible in the sense that if the number n + 1 is reduced, the theorem is no longer true for all sets X C R". However, the theorem can be sharpened when attention is restricted to special classes
of subsets of R", such as those which are connected (3.5) or have certain is said to be convexly connected symmetry properties (3.6). A set in such that the set misses H but provided there is no hyperplane H in intersects both of the open halfspaces determined by H. Results of Fenchel [1] may be stated as follows: is the union of n connected sets or is compact 3.5. SupPose the set X c and the union of n convexly connected sets. Then each Point of cony X is a convex combination of n or fewer Points of X. [1], the conclusion of 3.5 may fail even in As is noted by R2 if the set X is assumed merely to be convexly connected and to be bounded or closed but not compact. For Xc R2, let = H2(X) 111(X), the set of all points of cony X which do not lie in X or in any line segment joining two points of X. Then for various bounded convexly connected subsets X of R2, the set may have any finite cardinality and it may be countably or uncountably infinite (the latter proved by P. Erdös, using well-ordering). It is easy to produce closed convexly connected subsets X of R2 for which consists of one or two points. Danzer has given a complicated example in which consists of three points. Are there other possibilities? The following result of Fenchel [4] reduces to Carathéodory's theorem when m= 3.6. Suppose G is a group of linear isometries of onto itself, m is the dimension of the set M of all G-invariant points of and X is a subset of E" which is mapped into itself by each member of G. Then each point of M fl cony X lies in a set cony g Y for some set Y consisting of at most m + 1 points of X. It would be interesting to know just how much 3.3 can be sharpened when the set X is subjected to various restrictions. From 3.4 and 3.5 it is evident that cony X = !I(X) when X is as in 3.5 and j' n —' 1. We turn now to some results and problems which were inspired by Radon's [11, Stoelinga [1], Bunt [1], and
theorem. For each pair of natural numbers n and r, let f(n, r) denote the
smallest k such that every set of k points in
can be divided into r pairwise disjoint sets whose convex hulls have a common point. It follows from Radon's theorem that f(n, 2) = n + 2. The function f has been studied by R. Rado [6] and Birch [I], whose results are as follows: 3.7. For each n and r,f(n, r) (n + 1)r — with equality when a = I and
n=2.
LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE
118
The recursive inequality and determination of f(1, r) are due to Rado and the other results to Birch. Birch's argument employs Carathéodory's theorem, a fixedpoint theorem, and the result 2.8 on measures in R". Combining the recursive inequality with the fact that f(2, r) = 3r — 2, we see that always a bound which is sometimes' better than rn(n + 1)—n2 —
n
+ 1. Other minor
improvements are possible, but all known results are far from Birch's conjecture that perhaps always f(n, r) (n + 1)r — n. The above results imply 17, but the exact values of f(3,3) and 11 and 13 that 9 f(3, 4) are unknown. Several other problems are implicit in the paper of Rado [6], and a result related to one of these was obtained recently by Birch. For natural numbers n and r with r < n, let b(n, r) denote the smallest number k (if one exists) there is an r-dimensional flat which such that for each set of k points in contains r + 1 of the points and intersects the convex hull of the remaining points. Birch's result and its proof, not previously published, are as follows: 2r+2,b(7I,r) does not exist. 3.8. For n 2r+ 1,b(n,r)= n +2; for By are points of •, PROOF. Suppose first that n 2r + 1 and x1, Radon's theorem, there are real numbers a,,, not all zero but with zero sum, such that ax1 = 0. At least n — r + 1 of the a1's have the same sign, are all non-negative and so we may assume that the numbers a,, a > 0, we have at least one is positive. With s: tZ—r+1
=
.s42
(—a,/s)x,
1-4
this point lies simultaneously in the set cony {x1: 1 I n — r + 1) and in n + 2}. It follows that the smallest fiat containing {x,: n — r + 2 j b(n, r) n + 1. For the case n 2r + 2 it suffices to check that if c a A: = {a, a2, and
then A has the remarkable property that for any r + I points x0, ., of A misses the convex hull of (with r n/2 — 1), the r-flat through {x0, . . ., A
above was first given by Carathéodory 12]. It
established the following fact, later rediscovered by Gale 13] and Motzkin [31:
n, R" contains a convex polyhedron with k 3.9. For 2 m k and 2m vertices such that each m of these vertices determine a face of the polyhedron.
For further information on polyhedral graphs, see the papers by Gale [5], Griinbaum-Motzkin [2], and others listed by them. For an application of 3.9,
see 8.2+. We should mention also another interesting unsolved problem which involves choosing subsets of a finite set in such a way that certain convex hulls
HELLY'S THEOREM AND ITS RELATIVES
119
n < r, let h(n,r) denote the smallest number k such that in each set of k points in general position in R", there is an r-pointed set X which is convexly independent (no point of X lies in the convex hull of the remaining points of X). It is obvious that have empty intersection. For natural numbers 2
h(n, n + 1) = n + 1 and easy to verify that h(n, n + 2) = n + 3. Erdos-Szekeres
[1] mention a proof by E. Makai and P. Turan that h(2, 5) = 9, and they establish upper bounds for h(2, r) which are far from their conjecture that h(2, r) = + 1. In a later paper (Erdös-Szekeres [2)) they prove that h(2, r)
of his generalization of Helly's theorem is based on a De Santis's proof generalization of Radon's theorem which applies to sets of flats in a linear space. We shall state a slight extension of the latter result, but this requires further notation. For each finite or infinite sequence Ta of integers with 0 r2 ..., let e(ra) denote the smallest integer k for which the following is true: whenever FG is a sequence of flats in a real linear space, each
being of deficiency r, then the set of all indices i can be partitioned into
jr,) fl complementary sets I and J such that the intersection (cony (cony contains a flat of deficiency k. When no such k exists, define = oo• The following is a slight improvement of De Santis's generalization of Radon's theorem.
3.10. If m is the smallest integer for which = 00• this fails to apply,
< m, then e(ra) =
When
It may not be obvious that 3.10 is a generalization of Radon's theorem. To Each point may see that it is, consider a sequence x1, •, of points in be regarded as a flat of deficiency n, so the corresponding sequence r1, has always = n. When k n + 1, then m (in the statement of 3.10) is equal provided this is defined; otherwise E(Ta) to n + 1, and thus = It follows that when k n + 2 the set of indices (1, , k} can be partitioned into complementary sets I and J such that the intersection cony (xi: 1€ lJ fl cony : j JJ contains a flat of deficiency n; that is, the intersection is non.
empty.
By combining the ideas of 3.10 and 3.8, one may obtain further zations of Radon's theorem. However, these appear to be of only marginal interest. 4.
Generalizations of Helly's theorem. This section contains some generali-
zations and other relatives of Helly's theorem, all in except for a few supplementary comments. (See § 9 for generalizations in other settings.) At the end of the section we formulate some general problems which may serve as a guide to further research in this area. Division of material between § 3 and § 4 is somewhat artificial in view of the close relationships between the theorem of Helly on the one hand and those of Carathéodory and Radon on the other. Nevertheless, it seems to be justified in that the two different lines of investigation are different in spirit and lead to different generalizat ions.
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
120
Helly's theorem may be regarded as saying something about the "structure" of convex sets in RN for which = 0. Specifiof certain families . are compact, then cally, it says that if 2' is finite or all members of such there is a subfamily .7 consisting of at most n + 1 members of that = 0. This suggests the attempt to find theorems which say someof convex sets in R' for which thing about the structure of every family = 0, and which has Helly's theorem as a consequence. The known partial results in this direction are summarized in a separate report by Klee [6], since they seem more akin to the infinite-dimensional considerations.
How-
ever, the relevant papers are listed also in our bibiliography (Gale-Klee Lii, Karlin-Shapley [1], Klee [2; 4), Rado 151, Sandgren [1)).
of convex sets, Helly's theorem asserts As applied to certain families . The set {x) may the existence of a point x common to all members of be regarded, for j = 0, in any of the following six ways: as a
f-dimensional convex set which JIS contained Zfl} each member of .2'. f-dimensional flat Lintersects
(j + 1)-pointed set In each of the six cases, we may ask for conditions on a family which assure the existence of such sets corresponding to other values of j. Two of these questions are redundant, for there exists a f-dimensional convex set intersecting all members of if and only if there exists a f-dimensional flat intersecting all members of and (since the members of are convex) for j I there exists a (j + 1)-pointed set which is contained in all members of if and only if there exists a 1-dimensional convex set which is contained in all members of The remaining four questions have all led to generalizations of Helly's theorem. The most recent of these is the .
following theorem of Grünbaum [18):
41.
Let
g(n,O)=n+1,g(n,l)=2n,g(n,j)=2n—j for 1<j 0. Then there are positive numbers a,, , a with j n and members that all xeC. To deduce theorem from 4.5, let be a finite- family of compact convex sets in R", C a compact convex set confaining their union, and for each Ke and xe C let = p({x}, K). Then if the intersection is empty, the set of functions 0: = {%oR: Ke satisfies the hypotheses of 4.5 and hence there exist positive a1, - •, a- with j n such that fticOK,(X) > 0 for all x C. This implies that = 0 and Belly's theorem follows. A well-known topological theorem of Knaster-Kuratowski-Mazurkiewicz Li] has the following corollary, of which an elementary proof was given by Klee [ii: If j + 1 closed convex sets in have convex union and each j of them have a common point, then there is a point common to all. Using this fact as a lemma, Levi [2] proved:
46. Suppose
is a finite family of at least n closed convex sets in R" and has the following two properties: the union of any n + 1 members of .2 has connected complement in R"; each n members of have a common point.
Then there is a point common to all members of
-
This may be regarded as a generalization of Helly's theorem, since L41 is obviously implied (for n> 1) by the condition that the members of are bounded and convex and each n + 1 have a common point. The above lemma was extended in another way by Berge [2] and Ghouila-
Houri [1], though their results do not formally imply Belly's theorem.
than stating in full the result of Ghouila-Houri Ii],
we
Rather
state its two most
interesting corollaries as 4.7. Suppose C1, C,,, are closed convex sets in a topological linear space and each k of the sets have a common point, where 1 k < m. If U" c, is convex, then some k + 1 of the sets have a Point. If k = m — 1 and
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
124
fl' C = 0, Then each m — 1 of the sets have a common point in every closed set X such that X U U' C1 is convex. The second part of 4.8, describing the "hole" surrounded by the sets in question, may be compared with the description in 4.3+. The result 4.7 can also be approached by means of the Euler characteristic (Hadwiger [5], Klee [71).
For a family 5 of sets, let us define the Helly-number a(9)
to
be the
is a finite subfamily of 5 and smallest cardinal k such that whenever Helly's with card then 0 for all = n + 1 when is the family of all convex theorem asserts that The notion of Helly-number is especially appropriate for subsets of c families jr which are intersectional in the sense that E 9— for all
0 for each compact Xc E*, with 4(X) = 0 characterizing the convex sets and For example, GrUnbaum [10] defines a measure of noncon vex ity 4(X)
4(X) < oo if and only if X is the union of a finite family of pairwise disjoint compact convex sets. He proves 4.8.
For finite c
with 4(X)
e.
0, let
Then
e) is
denote the family of all compact sets Xc E" intersectional and has finite Helly-number.
Of course 0)) = n + 1. Griinbaum gives estimates for e)) and examples showing the impossibility of improvement in certain directions. The following result was proved by Motzkin [21: 4.9. Let d) denote the family of all varieties in (real) affine or projective n-space which are defined by one or more algebraic equations of degree Then ..4'(n, d) is intersectional and a(...,f(n, ii)) = (fl + d) In seeking Helly-type theorems for various non-intersectional families
one may encounter difficulties caused by the fact that the intersections of members of
can be structurally much more complicated than the individual
members. For such 5, it may happen that the Helly-number a(9) is infinite and that a more appropriate notion is the Helly-order a°(5), defined as the smallest cardinal k such that whenever is a finite subfamily of .5
ir.9' 5 for all .5' c .'? with card .9" In the first case, is an abstract complex whose vertex-domain is .9,
with
NJr
(Le.,
is
the vertex set of a simplex in N5) if and only
* 0. The problem is to characterize intrinsically the abstract complexes which (up to isomorphism) can be obtained in this way. In the second case, .., is a function on (1, --, to {O, l} such that A(11, "-, 1 if and only if 0. The problem is to characterize intrinsically the FiG arrays which can be obtained in this way. Of course the two problems are equivalent, but the different formulations may suggest different approaches. It seems that the ideas of Hadwiger [2; 5] might be useful here. Notice that the desired characterizations must depend on the dimension of if
the space, for with unrestricted dimension every intersection pattern is possible. Helly's theorem implies that for families in the complex NJJr is determined
HELLY'S THEOREM AND ITS RELATIVES
is determined by by its n.dimensional skeleton and the function A(FI, •-, having at most n + 1 different entries. its restriction to k-tuples (i1, .. •, the analogously Alternatively, one may consider in place of A(F,, ., ,I+1 to {O, ij, and ., k} . which is a function on {1, •, defined array n + 1. Further Fe's for j expresses only the incidence of j-tuples of the restrictions are imposed by the result of Hadwiger-Debrunner [2). However, the problem of general characterization appears to be very difficult and is not trivial even for the case of R'.' The one-dimensional case was encountered by Seymour Benzer [1; 2] in connection with a problem in genetics. He was concerned with those k x k for some k-tuple of intervals on matrices which have the form M2(F1, - •-, the line. Alternatively, one may ask for an intrinsic description of those graphs G which are representable by intervals—that is, which can be realized of linear as the 1-skeleton of a nerve N.9r for some finite family intervals. In this form, the problem was recently solved by LekkerkerkerBoland [1], whose result is as follows: •
-
4.12. A finite graph G is representable by intervals if and only if it satisfies the following two conditions:
(a) G does not contain an irreducible cycle with more than three edges; among any three vertices of G, at least one is directly connected to each path joining the other two.
They also characterize the representable graphs as those which fail to have subgraphs of certain types, and they describe some practical methods for deciding whether a given graph is representable by intervals. We turn now to an abstract description of some of the types of problems which arise in connection with Helly's theorem and its relatives. Suppose X is a given space and is a hereditary property of families of subsets of X. implies That is, is a class of families of subsets of X, and .5- c e
For each cardinal K, define the
by agreeing that
and card 5- 0, let
d
0:
HELLY'S THEOREM AND ITS RELATIVES
J(p, p'; d) = sup (rad, Y: Y c E, diamw Y
133
d}.
Existing literature treats only very special cases of the problem, and in particular is restricted to the case p = p'. Accordingly, we define J(p, d): = J(p, p; d). In this notation, Jung's theorem asserts that if p. is the (Euclidean) distance Santaló [51 describes the function J(p, d) in E*, then J(p,, d) = [n/(2n + (See also the where p is geodesic distance on the unit sphere S" in discussion of spherical convexity in 9.1.) The above formulation subsumes various problems concerning covering by positive homothets of a set in a linear space E. In particular, suppose C is a convex body in E with 0 e mt C, and for x, y e E let O:y
= inf {t
—
xetC).
Then p satisfies the triangle inequality arid is symmetric when C = —C. (For metric spaces with nonsymmetric distance, see Zaustinsky 11].) The function J(pc, d) is positively homogeneous in d and hence is determined by the value when E is n-dimensional of JCoc, 1). It is known that J(po, 1) n(n + and C = —C (see 6.4-6.5 below).
In the above setting, it is natural to discuss centers as well as radii of = rad, I. For sets, where a p-center of I is a point xe E for which distances pa,, such centers always exist in the finite-dimensional case, and under the Euclidean metric the center is unique (for bounded I) and lies in the closed convex hull of Y. The last condition is almost characteristic of the Euclidean metric, for Klee [5) has proved the following: For a normed linear space E, the following three assertions are equivalent: a° E is an inner-product space or is two-dimensional;
6.1
b° ,f a subset I of E lies in a cell of radius o''(C). Note that for a triangle S in R8, o'(S) 3>2= We turn now to Gallai's problem for families of homothets, where even less is known. The geometrical situation is so complicated that it seems impossible to find an equivalent covering problem. However, two different approaches by means of covering do lead to rough upper bounds. One possibility is to generalize the approach of Ungár and Szekeres mentioned earlier. For a convex body C in R', let ?(C) be the smallest number j which
is such that whenever Xc 1 and (x + a,C) fl C * 0 for each x€X, = {C) U {x + aRC: xe X), then .9 admits a f-partition (i.e., some f-pointed set intersects all members of $r). Then clearly Tz(HC) ?(C); and 2? C if 11± is the group of all homotheties in Further, with .9and
as described, a1 e (a1 + C) 11 C and yx: = a'(x — Zg) + zg, it is easily verified that + Ccx + a2C and + C) fl C * 0. Thus ?(C) may be defined alter.
natively in terms of families of translates of C. From this it follows, by reasoning analogous to that in 6.3, that ?(C) is equal to the covering number [C + (—C)/CJ. Thus we state 7.9.
r2(HC)
' Perhaps
?(C) = [C + (—C)ICJ.
this bound is already outdated by the method of Erdös-Rogers (11. (Cf. the
footnote to 7.10.)
HELLY'S THEOREM AND ITS RELATIVES
147
The principal, known results on the function 7 may be stated as follows (Danzer [3; 4], Grunbaum [9]): 7(C) 5' when C is centrally symmetric, (4n + 1)' in any case. 7.11. If C is a convex body in R2, then 7(C) 7 when C is strictly convex, while 7(C) 7 if an affinely regular hexagon A can be inscribed in C in such a way that C admits parallel supporting lines at opposite vertices of A.
7.10. For a convex body C in
while 7(C)
To show that 7(C) 5' when C = — C, one obtains a covering of 2C by first packing into 5C/2 as many translates of C12 as possible, and then expanding each of them by a factor 2. For general C, assume the centroid to be at the origin and let K: = C A (—C). Then C c nK by the result of Minkowski [1] and Radon [1) (or use 2.7) and hence C + (—C) c 2nK. Covering 2nK by translates of K leads to the inequality 7(C) (4n + 1)'. The first inequality in 7.11 follows from the possibility of arranging six translates of C so that they are all adjacent to C and are cyclically adjacent among themselves. For detailed proofs ot 7.10—7.11, as well as for relevant examples and values of rt(HC) for a few special C c R, see Danzer [2; 4]; for the case C = —C, see also Grtlnbaum [9]. Danzer [41 gives conditions on a family .9 in R' which imply the existence of a 7-pointed set intersecting the interior of each member
of .9.
For a Euclidean cell K, the idea of studying by means of 7(K) was extended by Danzer [3] to r,(K). With the aid of a general theorem on intersections of metric cells (stated in 9.9 below), he proved the following crucial lemma: 7.12.
Suppose 2 j
n + 1 and
is a family of Euclidean cells in E'
Let C0 be an intersection of j —
1 members of which has smallest (Euclidean) diameter among all such intersections, and suppose
with
KflC0*Ø (for all Then E' contains a flat F of dimension n + 2 — j such that F fl C0 is a Euclidean cell; diam C0 = diam (F fl C0) diam (F fl K) for all K€ (Ffl K) fl C0 0 for all K€ and (*) holds, then intersection with a suitable In other words, if flat wilt reduce the dimension by j — 2 and yield again a family of Euclidean ceLls in which each meets the smallest one. Consequently,
7.13.
• As pointed out by C.A. Rogers in a letter to the authors, these bounds may be replaced by 7(C) 3"O,, in the symmetric case and by in general, + where o,, : = n log n + n log log is+ 5n. Here 0,, is the density always obtainable by by translates of a convex body (Rogers (1]), while covering + 1)-' iS an estimate for the number vol ((1/3)C + (213)( — C))/vol (C) obtained by consideration of the (n + 1)-dimensional convex body associated with C (Rogers.Shephard 121).
148
LUDWIG DANZER, BRANKO GRONBAUM, AND VICTOR KLEE
It would be interesting to know whether, for 1 e family JY of Euclidean cells in E' with
i j — 2 and for every there exists a fiat F of
= (Ff1 K: Ke .2"I deficiency i in such that the family of intersections intersects and some smallest member of has property -' (or at least all members of In particular, what about the case i = j — 3? A second and simpler approach to the Gallai problem for families I-IC was recently developed by Danzer (as yet unpublished). Consider a family .9: = 1, and {x + : xE X} of homothets of C with Ce..9' and with always
For each k,.. can be split into k subfamilies, i k) e): = {x + aRC: xe X, (1 + < (1 + (1 then and a remainder Jr'(k, €) consisting of rather large cells. If .5 each of the k families €) can be pierced by ö't(C, 1 + e) or fewer points, while 9'(k, €) can be pierced by a rather small number p of points. (If C is smooth and (1 + is large enough, p n + 1.) It follows that Tj(HC) ko't(C, 1 + €) + p k[(1 + c)(J'o)C/Cl + p. suppose
0.
For large n this should yield much better results than the inequalities mentioned above, except when C = and I is close to n. In particular, this method
leads to the inequality k[(1 + €)(
—
+
1))'B"/B"] + RI + (1 +
where the best numerical results are obtained for (1 + e)i/n)(n + 1)'(j — 1 + (1 + Of course all the upper bounds for which have been mentioned here
are very rough. We know of no example which contradicts T,(HB*) n + 4—f and also of no C for which r,(TC) > n + 3—j. As to lower bounds for nothing is known for j > 2. *
*
*
*
*
*
*
*
*
In attempting to find values or upper bounds for the Gallai numbers and we were led to various covering problems. For general C and
n, difficulty is caused by the fact that the bodies to be covered by translates of C are themselves not much larger than C. For the special case of B", there is more hope of improvement. Recall from = o' t(C) 7.6 that whence (7.7) + 1XJ —
And by 7.9, 32(HC) ?(C) = [C + (—C)/CJ, whence
r,(HB")
[2B"/B"]
Thus we are interested in finding "good" coverings of rB" by translates of
especially for 1 a
C_(a)
1
+1
for
ff14
>
a
ir/6.
For small n and specific valifes of a, one may use particular coverings to
get upper bounds. For example, Danzer [3] shows whence f(B') 21 and 7.15. Ct(ff/6)=6, C2(ir/6)_20, and (See 7.9 — for definition of ?.) For larger n and arbitrary a one
71.
may instead estimate /ackings of radius
a12 by use of Blichfeldt's method (see Rankin [1]). This yields (Danzer [3]): 7.16.
0 such that whenever 9- and are families of homothets of C with 5 >and * 0, then a common point is obtained upon expanding all the members of about their centers by the factor c. The number is defined similarly for families of translates of C. Clearly 1 and 1 E0' are where Jo and the Jung constant and the expansion constant of 6. Is it possible to have when C is a Leichtweiss body (cf. 6.5+)? = E0? What is 1 9. Generalized convexity. The applicability and the intuitive appeal of convexity have led to a wide range of notions of "generalized convexity." For several of them, theorems related to Helly's were either a motive or a
by-product of the investigation. For others, it seems probable that no attempt has been made to find analogues of Helly's theorem. In order to encourage such attempts and to facilitate comparison of the various notions which have been considered, our discussion will include several generalizations of convexity which seem at present unrelated to Helly's theorem.
The usual procedure in defining a generalized convexity is to select a property of convex sets in R or E • which is either characteristic of convexity or essential in the proof of some important theorem about convex sets, and to
formulate that property or a suitable variant in other settings. Many properties of convex sets are useful for this purpose. We shall first describe in general terms the most important procedures which have been adopted, and then review' briefly some of the results obtained in specific cases. The usual definition of convexity in I?' can be generalized according to the following scheme. In a set X, a family of sets is given together with a function 11 which assigns to each Fs a family of subsets of X. A set K c X is called 'i-convex provided K contains at least one member of ij(F)
156
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
whenever,F c K and Fe .F. This is the most common form of generalization, and appears in most of the cases discussed below. Usually .9 is the family. of all two-pointed subsets of Xand Prenowitz [1), for example, studies convexity
in abstract geometries based on the notion of "join." For other possibilities see 9.7 below and Valentine [4), where one such variant is discussed and references to others are given. is an Another approach derives from the fact that every convex set in intersection of semispaces (Motzkin [11, Hammer 11; 2), Klee [41), and every
closed (resp. open] convex set is an intersection of closed [resp. open] halfof subsets of a set X is given, and a set Kc X is spaces. A family called
-convex provided
K is the intersection of a subfamily of
of this appear in 9.1, 9.3, and 9.9 below. See also Ghika [3).) The '-convexity can also be expressed in terms of "separation": a definitions of -convex if and only if for every x E X K there exists He set K c X is such that K c H and x e H. (For a generalization in this direction, see Ellis [1].) The family " may be defined in terms of a family of functions, or more directly we may be given a family 0 of real-valued functions on X and say that a set K c X is 0-convex provided for each XE X — K there exists 'p €0 such that ço(x) < inf çoK. This procedure is followed by Fan [1] in his generalization of the Krein-Milman theorem, and enters also in various notions of "regular convexity" (see Berge [1) and his references). is convex if and Another approach is based on the fact that a subset of (Instances
only if its intersection with each straight line is connected. Working in R' and using in place of the lines a two-parameter family of curves, Drandell [lj obtained interesting results under very weak assumptions. His paper also gives references for previously known results in this direction. (See Skornyakov [1] for a relevant result on curve families in R, and Valentine [3], Kuhn [I), Fáry [I], Kosiflski [1), and their references for some characterizations of convexity in terms of k-dimensional sections.) Of course one may generalize, in various ways, the underlying algebraic
structure in terms of which convexity is formulated. Many of the combinatorial results of this paper are valid in a linear space over an arbitrary ordered field. Monna [1] discusses convexity in spaces over nonarchimedean ordered fields. Other algebraic formulations are those of Rado 121 and Ghika [1; 2], the former being described briefly in 9.4 below.
One may ignore the surrounding linear space and seek a more intrinsic notion of barycentric calculus or convex
space.
This was done by Stone (1; 21,
H. Kneser [1], and Nef [I]. Their theories turn out to be embeddable in linear spaces, but might be of independent interest if reformulated to introduce topological as well as algebraic structure. For an open subset D of a metric space (M, p), let ö, and denote the inner and outer distance functions of D. These functions are defined on D and M respectively, with O,,(x): = p({x}, M-'- D) and vn(x): = p D is equivalent to that of the function — log ÔD and also to that of the function This suggests that for a family 0 of functions on subsets of a metric space (M, p), one could
HELLY'S THEOREM AND ITS RELATIVES
157
say that an open set K c M is #-convex provided —log Jj eG, or perhaps provided Vg
The notion of convexity in spaces of several oomplex vari-
ables may be approached in this way (see 9.11). However, a more useful idea suggested by these considerations is that one should try to generalize the sort of "convex structure" which is formed by the convex subsets of R' together with the convex or concave functions defined over them. This leads in one direction to the theory of complex convexity (9.11) and in another to the abstract minimum principle of Bauer [11, generalizing the Krein-Milman theorem.
Finally, we mention the fact that each new characterization of convexity in R' may lead to new generalized convexities, and that, conversely, the search for a suitable notion of convexity in a given setting may lead to discovery of new and useful properties of sets which are convex in the classical sense. For an interesting example, see R&dström's work ([1; 21 and 9.10 below)
in topological groups. We proceed now to discuss some specific "convexities," commencing with
those which appear to be most closely related to Helly's theorem. 9.1. Spherical convexity. Convexity on the n-dimensional sphere S" has been considered from various points of view, and is certainly the most significant of the generalized convexities so far as Helly's theorem is concerned. Several different definitions of spherical convexity have been studied, though not always with due regard to the limitations which they impose. For the sake of simplicity, we shall discuss only closed sets, though in most cases this restriction can be avoided. (S) A set Kc is strongly convex if it does not contain antipodal points and it contains, with each pair of its points, the small arc of the great circle determined by them. Equivalently, K (being closed) is strongly convex if it does not contain antipodal points and is an intersection of closed hemispheres. strongly convex K is an intersection of open hemispheres, and vice
versa.
(W) A set Kc S' is weakly convex if it contains, with each pair of its points, the small arc or a semicircular arc of a great circle determined by them. Equivalently, K is connected and is an intersection of closed hemispheres.
(R) A set K c S' is Robinson-convex if it contains, with each of its non-
antipodal points, the small arc of the great circle determined by them. Equivalently, K is an intersection of closed hemispheres. (H) A set K c S* is Horn-convex if it contains, with each pair of its nonantipodal points, at least one of the great circle arcs determined by them. Obviously, every strongly convex set is weakly convex; weak convexity implies Robinson-convexity, and that implies Horn-convexity. The strongly convex sets and the Robinson-convex sets form intersectional families. All strongly convex sets are contractible, as are all the Robinson- or weakly-convex sets except for the great rn-spheres c S". According to purpose, one or another definition of spherical convexity may be more appropriate. Horn-
158
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
convexity was introduced by Horn [1] in proving his generalization of Helly's theorem. Robinson-convexity was used by Robinson [2) in studying congruenceindices of spherical caps (see also Blumenthal [2]). Weakly convex sets were considered by Santaló [4] among others, though the definition at the beginning of his paper is different and includes only a subclass of the strongly convex sets.
An exhaustive bibliography of papers dealing with spherical convexity would be very extensive. Almost every notion and result on convexity in
can be extended to Sn, and in many cases the extension has been at least partly accomplished in connection with needs arising from other problems. As an example, we mention a little-known paper of Vigodsky [1] which proves (Vigodsky's main theorem was an analogue of Carathéodory's theorem for proved earlier by Fenchel [2).) It seems that Carathéodory's theorem and its
variants have not been generalized in full to the spherical case, though such an extension is surely possible. For certain types of problems, the treatment in S' is more satisfactory than that in R', due to the possibility of dualization in Sn. For example, the of Jung's theorem [11 on circumspheres and Steinhagen's [1] analogues in on inspheres are dual aspects of the same result (Santaló [51). On the other hand, some results in have no Euclidean analogue. An example of this kind is furnished by the difference between the following two relatives of Jung's theorem, in which connectedness plays an essential role though it is irrelevant in Rn: If a compact subset of has diameter less than arc cos (n + 1)', it lies in a closed hemisphere; a compact subset whose diameter is equal to arc cos (n + 1)-' need not lie in any hemisphere (MolnIr [4]). If a compact connected subset of sn has diameter arc cos n1, it lies in a closed hemisphere (Grunbaum [14]).
Due to the close and obvious relationship between spherical convexity in
many results can be interpreted in both the and convex cones in spherical and the Euclidean setting. This fact has been used repeatedly (e.g.,
s"
Motzkin [1), Robinson [11, Horn [1]).
For spherical analogues of Helly's theorem, the simplest approach seems to be through Helly's topological theorem. From this it follows that the family of all homology cells in n + 2. Alternatively, the result has
may be stated as follows: A family .9 of homology cells in
has non-
empty intersection provided each uniofi of n + 2 members of ..9 is different from sn and each intersection of n + 1 or fewer members of is a homology cell. This applies, in particular, to the case of strongly sets considered by Molnár [4]. Similar corollaries may be derived for the other types of spherical convexity, but their formulation is somewhat more complicated due to the absence of intersectionality, or contractibility, or both (see Horn [11, Karlin-Shapley [1], Griinbaum [14]).
Other results of Helly-type for s" may easily be derived from results on convex sets in For example, the case j = 1 of Theorem 4.1 (essentially contained in Steinitz's result 3.2) may be reformulated as the following fact, first stated by Robinson [2]: If is a family of at least 2n +2 Robinson-
HELLY'S THEOREM AND ITS RELATIVES
159
and each 2n +2 members of .9- have a common point, then r.9- * 0. This and related results were employed by Blumenthal [21 in connection with linear inequalities. For some intersection and covering theorems a question involving hemispheres, see Hadwiger [4] and Blumenthal [2; raised by Hadwiger is answered by GrUnbaum [9]. Horn-convex sets have been studied by Horn [1J and Vincensini [3; 4; 5]. Horn shows that if 1 k n + 1 and 9- is a family of at least k Hornsuch that each k members of ..9 have a common point, convex sets in lies in a great (n — k + 1)-sphere which then every great (n — k)-sphere in convex sets in
intersects each member of .9 (cf.
4.3).
For k =
n + 1,
this asserts the
existence of a point z such that every member of F includes z or its antipode. 9.2. Projectlve convexity. A set K in the n-dimensional (real) pro jective space P' is called convex provided it contains, with each pair of its points, exactly one of the two segments determined by these points. Such sets have
been considered repeatedly since their introduction by Steinitz [1] and VeblenYoung [1], though most authors have discussed only the equivalence of various definitions. (See de Groot-de Vries [11 and other papers listed by them.) The
projectively convex sets are contractible but do not form an intersectional family. The intersection of k + 1 convex sets in may have up to components, each of which is projectively convex (Motzkin [1]). On the other
hand, if each two of the sets have convex intersection, then so has the entire family. Thus the following is an immediate consequence of Helly's topological is such theorem: If a family 9 of at least n + 1 closed convex sets in that
of .9- have convex intersection and each n + 1 members intersection, then x.9 * 0 (Griinbaum [11]). let which are the k) denote the class of all subsets of
each two members
have
For k
1,
union of k or fewer pairwise disjoint closed convex sets. It seems probable that k) is of finite Helly-order, but apparently this has not been established and surely the exact value even of 2)) is unknown (cf. 4.11). For other results related to projective convexity, including some on common transversals, see Fenchel [3], Kuiper [1], de Groot-de Vries [1], Schweppe [1J, Gaddum [1], Marchaud [1], Grunbaum [11] and Hare-Gaddum [1]. 9.3. LevI's convexity. Like that of Rado discussed in 9.4, this notion was motivated primarily by Helly's theorem. Levi [1] considers a family of subsets of a set X, and assumes: (I)
for all
For each set Y lying in some member of the is defined as the intersection of all members of 'if' which contain Y. The second axiom is (ha) Every (n + 2)-pointed subset of a member of contains two disjoint sets whose have a common point. Generalizing Radon's proof [21, Levi deduces from the above axioms that if
.9- is a finite family of at least n + 1 members of and each n + 1 members of 9- have a common point, then * 0. As corollaries he obtains Helly's theorem on convex sets in R", its extension to n-dimensional
160
LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE
geometries satisfying Hubert's axioms of incidence and order (Hubert [11), and an intersection theorem in free abelian groups with n generators. (In is the family of all subgroups, the neutral element being the last case, omitted from each.)
It would be of interest to study, in a system assumed to satisfy (I) and perhaps other simple "axioms of convexity," the inter-relationships of Radon's property as expressed by Helly's property as stated above, and Carathéodory's property expressed as follows: (ilL1) Whenever a point p lies in the
of a set Y, then p is in
the
so,neai-nwst-(n + 1)-pointed
subset of Y. 9.4. Rado'. convexity. Rado [21 considers an abelian group A under the action of a commutative ring of operators, and sets forth conditions on A and which assure the validity of Hefty's theorem (with a proper
interpretation of "convex set"). His reasoning yields not only Helly's theorem on convex sets in R*, but also a generalization of a theorem of Stieltjes Eli on arithmetic progressions. With respect to a given coordinatization of a lattice is the set of all points of the form x0 + where the xj's are n + 1 given points with integral coordinates and the ai's are-arbitrary integers. Rado proves that if ..9 is a finite family of at least n + 1 lattices in such that each n + 1 of them have a common point, then * 0. 9.5. Ryperconvexity. If K is a compact convex set in R*, a set A c R' is K-convex (or hyperconvex with respect to K) provided A contains, with each pair of its points, the intersection of all translates of K which contain those points. K-convex sets form an intersectional family and the connected Kconvex sets are intersections of translates of K. K-convex sets need not be connected, but attention is usually restricted to their connected components. The study of hyperconvexity was initiated by Mayer [1], and a complete set of references can be found through Pasqualini [1), Blanc (I], and Santaló [6). Most of the papers treat only the planar case, and under restrictions on K. They deal with support properties, equivalent definitions, hyperconvex hulls, extremal problems, etc. 9.6. Quasiconvexlty. Let A be a subset of [0, 1). A set Kc is A-convex (or quasiconvex with rvspect to 4) provided Ax + (1 — 2)y
K whenever A e 4
and x,y e K. This notion was studied by Green and Gustin El), who give references for the previously considered special cases (such as A = (1/2)). A detailed study of even more general concepts is the paper of Motzkin [4J. 9.7. Three-poInt convexity. A set K in R" is 3-Point convex if it contains,
with each three of its points, at least one of the three segments determined by these points. Valentine [4; 5] has made an interesting study of this property, and a more general notion has been formulated by Allen [1]. Hare and Gaddum [1] discovered a connection between 3-point convexity and projective convexity. It seems likely that the family of all 3-point convex subsets of is of finite Hefty-order, but this has not been determined. (See also Valentine [6; 7].)
HELLY'S THEOREM AND ITS RELATIVES 9.8.
161
A set K in a partially ordered space is order-convex
K whenever x, z K with x 0, if it is to be must have PH as the corresponding eigenspace. But f(A + H) has only one eigenvalue >0 and fails to satisfy this condition. EZAurIz 5. The absolute value function v is not matrix-convex on (—2,2) (evidently, then, not on any interval including 0). Paoor. Let
A=('
r>0.
Then
v (f A + ÷ B) = 4
(r
4- v(A) +
÷ v(B) =
but for small i the latter expression has a smaller entry in the lower right
corner than does the former. i By positive homogeneity, this is the same as saying the "triangle inequality" V(A) + v(B) v(A + B) does not hold; indeed, the same example refutes any "weakened triangle inequality" of the
200
CHANDLER DAVIS
form v(A) + v(B) pv(A + B) 121]. EXAMPLE 6. The square function is operator-convex. This is a familiar computation: ((1 — A)A + (1 — 2)A' + AB2 is equiva-
lent to 0
(1 — A)A(A — B)2. EXAMPLE 7. The square function
P is a projection, II P11
satisfies the Sherman condition. 1, so ((PAP)2x, x) =
(PA'Px, x).
The inverse function satisfies the Sherman condition on (0, 00);
EXAMPLE 8.
that is, for invertible A
0 and projection F, (PAP)-' PA1P, the left side
being defined on PH. PROOF. Clearly PAP is invertible on PH. For any xe PH, define H, z = (PAP)'x PH. Now the object is to prove that, of the two positive numbers ((PAPY'x, x) = (z, x) anr (PA'Px; x) = (y, x), the second is the first.
But 0
so
(A(y
—
z), y — z)
= (Ay, y) — (Ay, z) — (z, Ay) ÷ (Az, Pz) = (x, y) — (x, z) — (z, x) + (x, z).
1. N. Aronszajn, La théorie des noyaux riproduiaanta et sea applications. 1, Proc. Cambridge Philos. Soc. 39 (1943), 133-153. 2. J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58-71. 3. Yu. L. Daleckil, Integrirovanie i differencirovanie ftnkcit èrmitovyh operatorov, of parametra, Uspehi Mat. Nauk 12 (1957), no. 1(73), 182-186. Translation; Amer. Math. Soc. Transl. (2) 16 (1960), 396-400. 4. C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957), 42-44. 5. , AU convex invariant functions of hermitian matrices, Arch. Math. 8 (1957), 276—278.
6.
•
Various averaging operations onto subaigebras, Uhinois J. Math. 3(1959),
538-553.
7. , Operator-valued entropy of a quantum mechanical measurement, Proc. Japan Acad. 37 (1961), 533-538. 8. T. Kato, Notes on seine inequalities for Linear operators, Math. Ann. 125 (1952), 208-212.
9. A. Korányi, Note on the theory of monotone operator functions, Acta Sci. Math. Szeged 16 (1955), 241-245. 10. , On a theorem of and its connections with read vents of esjfad-
joint transforinattons, Acta Sd. Math. Szeged 17 (1956), 63-70. 11. , On some classes of anal ytic funot ions of several carialijes, Trans. Amer. Math. Soc. 101 (1961), 521-554. 12. F. Kraus, ()ber konveze Matrixfunktionen, Math. Z. 41 (1936), 18-42. 13. C. Loewner, Advanced matrix theory, Lecture Notes, Stanford Univ., 1957. 14. K. Löwner, tfber monotone Matrixfuhktionen, Math. Z. 38 (1934), 177-216. 15. C. C. MacDuffee, The theory of matrices, Springer, Berlin, 1933.
16. M. Marcus, Convex functions of quadratic form., Duke Math. 3. 24 (1957),
321-326.
NOTIONS GENERALIZING CONVEXITY
201
17. M. Nakamura, M. Takesaki and H. Umegaki, A remark on the expectations of operator algebra., Ködai Math. Scm. Rep. 12 (1960), 82-90. 18. M. Nakamura and H. Umegaki, A iscte on the entropy for operator algebras, Proc. Japan Acad. 37 (1961), 149-154. 19. A. Ostrowski, Sur queique. applications dee functions convenes et concaves au eons de I. Schur, J. Math. Pures AppI. (9) 31 (1952), 253-292. 20. 1. Schur, Obey cuss Kiasse ven Mittetbildungen mit Anwendungen auf die minantentheori., S.-B. Berlin. Math. Ges. 22 (1923), 9-20. 21. W. F. Stinespring, Problem 4863. Amer. Math. Monthly 66 (1959), 728. 22. B. Sz.-Nagy, Proiongements des transformations de i'eapace de Hubert, qui sortent de cet espace, Appendice au livre "Lecons d'analyse fonctionelle" par F. Riesz et B. Sz.Nagy, Budapest, 1955. 23. B. Sz.-Nagy and A. Korányi, Operatortheoretiache Behandlung und VeraUgeineinerung cix.. Probiemkrei.ea in dcv hemplexen Funktionentheorie, Acta Math. 100 (1958), 171-202. MATHEMATICAL REVIEWS
SOME NEAR-SPHERICITY RESULTS BY
ARYEH DVORETZKY' 1. The sphere occupies a very special position in geometry and especially in the theory of convex bodies. The relevant results are too numerous to quote, yet the subject is far from exhausted and new aspects continue to be
The purpose of this communication is to discuss one such recently discovered new facet. The results discussed here will center around the theorems reproduced in the next section. Since their proof is rather lengthy, it will not be reproduced and the reader is referred to [4]. Some other results discovered.
from [4] will also be quoted. On the other hand, new corollaries of these results will be drawn. In § 5 we will prove a new result on the projections of a cube, which is a counterpart to a result given in [4] on the sections of a cube but is more precise and, also, easier to prove. We shall close with a discussion of some open questions and also state a new result on the convergence of series in Fréchet spaces, which is derived from near-orthogonality—
a rather poor relation of near-sphericity, but which is more widespread. Except for the last section we shall deal exclusively with Euclidean spaces (see [3J for further applications to functional analysis). Throughout the paper E* will denote n-dimensional Euclidean space, its points will be denoted by x = (x1, ..., X,,) in orthonormal coordinates. BW will 1} of E'. C' will represent the stand for the unit ball {x; + - -. + unit cube {x; —1 1 (i = 1, -, ,s)} while will denote the unit octahedron Cx; I xjI + -. + 1) of E'. By a subspace Ek of E' (1 k n) we shall always mean a subapace through the origin. The term symmetric will always signify symmetry with respect to the origin, thus a set C c E0 is symmetric if and only if C = —C. A set C is said to be a convex body in it it is a compact convex set with nonempty interior. I
To facilitate the statement of our result we give the following definition. DEFnnnoN. A convex set C in a Euclidean space is called spherical to within s (0 < e 1) if there exist in the flat space generated by C two concentric balls B1 and B, of radii (1 — e)r and r, respectively, such that B1 c C c B,. The g. 1. b. of the t having the above Property is called the asphericity of C 2.
and denoted by a(C).
If C is a compact set then there exist concentric balls of radii (1 — and 1
a(C))r
r, the one contained in and the other containing C. If P is the common The
research reported in this paper has been sponsored in part by the Air Force
Office of Scientific of the Air Research and Development Command USAF through its European Office. 203
ARYEH DVORETZKY
204
center of these balls then 1 — a(C) is simply the ratio of the distances from P to the nearest and farthest points on the boundary of C. in case C has a center of symmetry, the point P must coincide with it. Obviously a(C) 1 for a compact set C if and only if C is a ball. r(C) is a continuous functional of C in the usual Blaschke topology of compact sets in E". The main result of [41 is THEOREM 1. Given e (0 < e < 1) and a positive integer k, there exists an integer N = N(k; e) such that if C is any convex body in E", n N, then there is a subs pace E" for which a(C
< c.
A suitable N(k, e) is given by exp
log k).
The crucial fact is that N(k, e) is a function of k and e alone, thus giving a result of a uniform nature for all convex symmetric bodies. Since a is a continuous functional it is obvious that if (1) holds for a certain E" it will hold also for all E" sufficiently "near" it. It is natural to ask for some measure-theoretic statement about the Ek satisfying (1). The obvious measure to use is one which is preserved under orthogonal transformations. According to A. Weil's results on Haar measures in homogeneous spaces [9], there exists a unique (up to a numerical factor) regular measure of this kind. We shall denote this measure, normalized so that it becomes a probability
measure, i.e., so that the measure of the set of all E" c E" is 1, by jg,,,i,. Explicit expressions for this measure may be found in works on integral geometry; these, however, need not concern us here. It is very easy to see that2 inf .a,,,k{Ek; a(C n < e} = 0 for every n, £ > 0 and k> 1 when the jnf is taken with respect to all symmetric convex C c E'. To see this it is sufficient to consider ellipsoids with all but one principal axes equal and the remaining one very large. Thus to obtain significant results of this nature some suitable restrictions must be imposed on C. A result of the type urn inf p,,,tc{E"; a(C n Ek) < s} = 1 (2) would be interesting already when
denotes the class, say, of symmetric
i.e., C which lie between two concentric convex bodies C with a(C) < A < balls with fixed ratio of radii. However, much more is true; not only (2) holds for a much wider class of convex bodies, but the result can be uniformized. Indeed, it was proved in [41 that the following theorem holds. THEOREM 2. Given e (0 < s < 1), 8 (0 < 6 < 1), an integer k> 1 and a function g(n) defined on the positive integers and satisfying (3)
g(n)
1 (n =
1,
2, ...),
urn
there exists an integer N = N(k; €, 6; g) such that 2
(Ek; . . .} denotes the set of Ek possessing the property
SOME NEAR-SPHERICITY RESULTS
a(C
(4)
holds
for all n
fl
Ek) < e} >
1
205
ö
N and all symmetric convex bodies C in E" satisfying
C cg(n)C". (We recall that B" and C" are the unit ball and unit cube of E", respectively.) Though we do not reproduce the proof of these theorems, it should perhaps be remarked that no direct proof of Theorem 1 is known and that in [3] and [4] it Is derived from a weak version of Theorem 2 (with g(n) = const). We would like also to note that the proof proceeds in two main stages: first it is shown that for C satisfying (5) there are "many" directions for which the of C are nearly equal, then integral distances from the origin to the geometric methods are used to derive (4). 3. So far we have discussed the near sphericity of sections, let us now turn to that of orthogonal projections. Let us denote by Cl Ek the orthogonal projection of a set C in E" on (1 k n). If C is a closed convex set containing the origin and e is a unit B"
vector then the projection on the ray determined by e is the segment [0, H(e)e],
where H(e) is the support function associated with C (for this and all other assertions about the general theory of convex bodies used here see [1]). Since if C is a symmetric convex body, the union of all such segments is again such a body, it follows immediately from Theorem I and the fact that if H(e) r that we have for all ee E& then rB" c Cl THEOREM 3.
Given
(0
0
for t * t17i = 1, •-, m.
Let p(t) be the polynomial defined by
+1k". Clearly p(t) is non-negative and vanishes only for I = I = 1, - - •, ,n. Letting ", fl2rn-i, 1) it is clear from (3) that b and satisfy (2) as required. We shall call a polytope of the type given above a cyclic Polytope and we
b = (Ps, /3k,
-
proceed to describe the face structure of such polytopes. Our first result applies not only to cyclic but to any neighborly polytope. The Points of an rn-neighborly set S in 2m-space are in general
THEoREM 2.
Position. PRooF. Suppose some set R of 2m + 1 points of S lies in a hyperplane. The set R is rn-neighborly since S is, but R would then be an rn-neighborly set in a space of dimension 2m — 1 and as proved in [1, § 3], such a set can contain at most 2m points, giving a contradiction.
The faces of a neighborly Polytope are simplexes.
We shall next determine exactly when a set of vertices of a cyclic polytope spans a 2m-face. For this purpose it is convenient to assume the numbers
occur in increasing order so that the points 01 , 02, •, 0,, occur in order on the moment curve. Let R be any subset of S containing 2m points. THEOREM 3. The Points of R are neighbors and only if (3) for any two Points 0,, 0 not in R there are an even number of points of and 0,. R between Let R = -. -, We consider the polynomial 2w,
P(t)
Letting b = (SB,,
-
•, thm)
fl(t
— ti,.)
=
it is clear as in Theorem 1 that the hyperplane
through R is given by the equation
NEIGHBORLY AND CYCLIC POLYTOPES
+ Pa = 0
b
P(11) + and this hyperplane will support S if and only if the numbers not in R. Suppose is not in R and p(t,) is are all of the same sign for positive (negative). This means from (4) that there are an even (odd) number But if 0 is also not in R then there will be an greater than of roots, even (odd) number of roots, 1,,, greater than I if and only if there are an even number of these roots between and tj which is precisely the condition in the statement of the theorem. There is a simple schematic representation to illustrate Theorem 3. Imagine
a set of n beads strung on a circular string and suppose 2m of them are black and the rest white. The black beads will then correspond to a face of S if in this necklace they occur in intervals of even length. It is clear from Theorem 3 that all cyclic polytopes in 2rn-space having the same number of vertices are combinatorially isomorphic, that is, there is a one-one correspondence between vertices which induces a one-one correspondence between faces. Motzkin has raised the question as to whether every rn-neigh-
borly set in 2m-space is combinatorially equivalent to a cyclic polytope. He indicated an approach to this problem which yields a partial result to be taken
up in the next section. We conclude the present section by computing the number of (2m — 1)-faces of the cyclic polytopes. THEOREM 4. The number of (2m — in 2m-space is given by the formula
1)-faces
of a cyclic Polytope with n vertices
PROOF. (We assume, of course, that n > 2m.) Let R be a face of S and let
1, Stein-
haus' proof applies as well in higher dimensions.
BRANKO GRUNBAUM
250
Among problems related to these theorems and generalizing the elementary we mention some raised in Griinbaum [5]: If K c E' is a convex results on
body, n 3, does there exist a point x e mt K which is the centroid of at least n + 1 different sections of K? (For an affirmative answer to this problem, see § 6.2.) What properties does the set of all such points x have? Does it contain the centroid of K?
The above results of Steinhaus are true also if, instead of centroids of sections, other continuous point-functions of convex sets are considered (such as surface-area centroids, centers of minimal circumellipsoids or circumspheres, etc. (see Grünbaum (5], Steinhaus [21)), and with respect to each such pointfunction the foregoing questions may be raised. (ii) Neumann [1] considered also certain other functionals, related to F1, which are easily modified in such a way as to become measures of symmetry. Following his notations, let p1, 0 i 2m, be endpoints of chords of and
all passing through a point xeK, and such that p' is etc. (See Figure 11 for m = 2.) Let = mm LIP.
between q,,,+1 — xIII p,
—
I
FIGURE ii
over all choices of p1 , q1 satisfying the above conditions, and let = max XE K}. Then m p2m+i(K) m + 1/2, where ptN+I(K) = m charac-
terizes triangles, whire
= m + 1/2 if and only if K has a center of
symmetry. For m 1 this result can easily be obtained from the properties of F1 (Neumann derived F1(K) 1/2 from p5(K) 1); it seems also possible to derive the estimate p2m+t(K) rn from 1/2. (iii) Similarly defined are the following measures of symmetry. For KE W' let x1, be points of hdKand let Let y, be
the other endpoint of the chord of K determined by x and x1, and let
c(x)=min
1
+1
—
xI
MEASURES OF SYMMETRY FOR CONVEX SETS
for all families {x1) with x E cony
Let G(K) = max {c(x): XE K), ir(K) =
K). From the results on F1 it follows that 1/n a(K) 1 and = 1 if and only if K ir(K) 1. It is obvious that o(K) = 1 or has a center of symmetry, and that the lower bound can be attained, if at max {lr(x): x
= all, only by the simplex 4". For ir it was proved recently that (Berkes [1], Schopp [1]); the assertion u(4") = 1/n is also easily established. 1. For each point p of a convex each hyperplane H with p e H, let f(H, p) be the ratio, not exceeding 1, of the volumes of the two parts of K determined by H. We define 1(P) = mm {f(H, p): Hsp) and F2(K) max {f(p): p K).
6.2.
WINTERNITZ' MEASURE OF SYMMETRY.
body Ke sr,
and
This measure of symmetry, or rather the derived measure F(K), was studied by different authors (see below); the bounds for F2(K) are the same as those for F(K). The only F2-extremal bodies are the simplices (Grünbaum It is easy to establish that the level surfaces of F2 are convex, and that there exists a critical point for each K€ Se" (Hammer [4]; also Süss [31 for n = 2). Applying a variant of Helly's theorem (see, e.g., Ortinbaum [6, Lemma 3]) it follows that there exist n + 1 or more hyperplanes H through the critical point '(K) of K each of which divides the volume of K in the ratio F2(K). Since for each such H the point is centroid of H n K (see l3laschke [3] for n = 3; the proof applies to all n), an affirmative answer to one of the problems mentioned on p. 250 follows. The property of F2 most difficult to establish is that F2(K) = 1 only if K has a center of symmetry. Though this fact is immediate for n = 2, the known proofs for = 3 (Funk [1], see Blaschke [5, p. 250], Bonnesen-Fenchel 11, P. 25)) use integral equations, or spherical harmonics. (The author regrets the historically incorrect statement made in this connection at the end of Gilinbaum [3].) Although it is possible to extend these methods to > 3,. no
[3], Hammer [4]).
such proof seems to have been published. Worth mentioning In this connection
is the interesting result of Ungar (1].' 2. The following results are known on affine-invariant measures derived from F2.
(i) The derived measure fl based on the centroid g = g(K). The history of bounds for F(K) is quite interesting. The case KE was first investigated by A. Winternitz, who established the inequality 4/5 and the fact that triangles are the only extremal bodies. Winternitz's results were published in 1923 in Blaschke's book [5, pp. 54-55); they seem to have been unnoticed and were rediscovered independently by Lavrent'ev-Lyusternik [1] (in 1935), Neumann [2] (in 1945), Yaglom-Boltyanskii [1) (in 1951), Ehrhart (1) (in 1955), and Newman [1] (in 1958). They were ascribed to Winternitz (but
without date or reference) in the German translation of Yaglom-Boltyanskii Li] (in 1956).
The inequality F(K) n"J[(n + 1)' — n"] for Ke .Q' was conjectured by Ehrhart 11) and established by Ehrhart [21 for n = 3 and by Grünbaum [31 and Hammer (41 for all n. The bound is attained if and only if K is a pyramid on any (n — 1)-dimensional convex base. See Note 4 on p. 264.
BRANKO GRONBAUM
252
(ii) Eggleston (2] (reproduced in (3] and in [51) proved for each six-partite point s of K the inequality = f(s) 4/5. From this and the uniqueness of F2-critical points follows at once the uniqueness of the sixpartite points for triangles (proved by a different method in Eggleston [2)). 3. Generalization of F1 to mass-distributions. Given any distribution p of in complete positive masses in E', with p(E*) < 00, one may define analogy to the definition of F2(K), except for substituting the mass contained in the half-spaces for the volumes used in the definition of F1(K). Obviously, F2(p) is not a measure of symmetry since F2(p) = 1 is possible for essentially noncentral mass distributions. On the other hand, F1(p) 1/n for any massFor n 2 this result was first proved by Neumann [2], distribution p in and rediscovered by [1] and Newman [1]. For arbitrary n the inequality was established by Rado [2], Birch (1], Grilnbaum [3];? see also Danzer-Grtinbaum-Klee [1]. The bound 1/n is obviously the best possible. A characterization of distributions for which it is attained is given in GrUnbaum
For a discussion of F1 in the plane, and some related notions, see
[31.
Neumann [31. 6.3.
Tnn SURFACE-AREA ANALOGUE OP
MPASURE.
Let F3(K), for
Ke sr, be defined in the same way as F2(K) except that f(H, P) is the ratio in which the surface-area of K is divided by H. From the results on F2(u) it follows at once that F3(K)> 1/n for Ke SI". Nothing more is known about the functional F8(K) for Ke sr with n 3. In particular, it is not known whether F3(K) = 1 only if K has a center. For n 2, it is easy to establish that F(K) < 1 if K does not have a center. Neumann [2] conjectured that the lower bound of F1(K) for K€ 512 (i/T — 1)/2; elementary computations show that the lower bound may not exceed (VT — 1)/2, since F2(K) comes arbitrarily close to this number if K is a sufficiently narrow isosceles triangle. Ehrhart [1] mentions that F3'(K)> 1/3k for Ke 512, but the proof seems not to have been published.
K
FIGURE 12
I
See Note 10 on p. 264. See Note 5 on p. 264. See Note 6 on p. 264.
MEASURES OF SYMMETRY FOR CONVEX SETS SYMMETRY. 1. Let p be a point of the interior 6.4. Two MEASURES of a convex body Ke r, H a hyperplane through p, H1 and H2 hyperplanes parallel and equidistant from H, and H1 supporting K. If K'(H3) denotes the part of K contained between H1 and H2, let , p) = V(K'(H1))I V(K) and = mm {f(H1 , p): H1 supporting K). We define F4(K) = max (1(p): pe K). Since F4(K) = 1 implies Fi(K) = 1, it follows that F4(K) = 1 iff K ha8 a center. that f4(K) 8/9 Fvom the known resultsftbotjj F:(K) (see below) it of for K€ with triangles as only extremal bodies. The exa4
F4(K), for Ke r, with n 3, are not known; we conjecture that F4(K) l)/(n + 1)]' for Ke R', with simplices as the only extremal bodies. The derived measure F41(K) was consideIed in Asplund-Grosswald-Grtinbaum for Ke it" cannot be determined in closed of 111. The lower bound 1 — [(n —
form for n 3, but may be computed_-numerically (e.g., a3 = 0.8746 "); the extremal bodies are pyramids on arbitrary (n — 1)-dimensional convex bases,
truncated in a suitable ratio (depending on n; e.g., 0.2-•- for n = 3). For the plane, a1 and the only extremal bodies are triangles. It is of some interest to note thiL denoting by 4% the n-dimensional simplex, we have =1— 0.857 ... = 0.865Let K be a convex body in E", p an interior point of K, and H1, H1 a pair n K, of parallel supporting hyperplanes of K. For each pair of points 1x1, x1] and containing P. let C(x3, x1) be the chord of K paralleito the 2.
FIGURE 13
H' — and
f(p) =
mm
(length C(x1, x3)
£7
length [x1 , x2)
(1(1', He): H1,
supporting
K).
—1 S—i,
We
define
F5(K) =
max {f(p): pe mt K).
The assertion "F1(K) = 1 if and only if K has a center" is equivalent to Theorem 1 of Hammer [3). It is not hard to verify that F5(K) satisfies the superminimality condition; therefore it is immediate that F5(K) 2/3 for Ke.Q2, with equality only for triangles.10 The lower bounds of F1 in higher 10
See Note 7 on p. 264. See Note 6 on p. 264.
BRANKO GRUNBAUM
254
dimensions seem not to have been determined; it may be conjectured that simplices are the extremal bodies (this would imply that for n = 3, F5(K) 1/2). For the derived measure F.' it is easy to show, using methods similar to those applied in the proofs of F,'(K) 4/5 for KG that FT(K) 2/3 for Ke The bounds of F1(K) for Xe r are not known for n 3. 3.
Some problems on affine diameters. If K is a convex body in E', a
chord C of K is said to be an affine diameter of K if and only if there exists a pair of (different) parallel supporting hyperplanes of K, each containing one of the endpoints of C. (The segments [x1, x5] used in the definition of F5 are affine diameters.) It seems to be well known that each point of K belongs to at least one affine diameter. This result was announced by Hammer (3], but his proof was not published. It may be easily derived from Theorem 1 of Grunbaum 15] and the remark following it. (It is also easy to establish that for each direction there is an affine diameter parallel to it.) The following
problems on affine diameters are similar to those on centroids of sections mentioned on p. 249. (For generalizations of the above results, and partial answers to the problems below, see Kosiflski 12; 3].)
For a convex body K in E*, does there exist at least one point contained in at least n + 1 different affine diameters? In particular, is the centroid of K such a point? Does each K have either continuum many points each belonging to more than one affine diameter, or a point belonging to continuum many affine diameters?
For n = 2 the affirmative answer to the last question is trivial. The answer to the first two is also affirmative; it is not hard to show that both the centroid of K and the F1-critical point of K belong to at least three different affine diameters. 6.5. and a
MEASURE OF SYMMETRY.
1.
For a convex body Ke r
point peK let K(p) = Kn (2/' — K) be the maximal subset of K with center at p. We define to be the ratio of the volumes of K(p) and K, i.e., = V(K(p))/ V(K); the Kovner-Besicovitch measure of symmetry is F.(K) = max Pc K). The symmetry measure F. has been the object of many investigations. It seems that the first published proof of F,,(K) 2/3 for is to be found in Lavrent'ev-Lyusternik [1, pp. 358-367), where it is credited to S. S. Kovner. Independently, simpler proofs of this inequality were given by Besicovitch [1], by Fáry (1), and, for derived measures, by other authors (see below). Fáry [1] and others also noted that F,(K) = 2/3 characterizes triangles. Very few results are known on F. in higher dimensions. Bielecki and Radziszewski [1] proved that every convex body Ke
contains a parallelepiped
P with V(P) (2/9) V(K). It follows that F.(K) 2/9 for Ke Ffiry and Rédei (1] proved that every KeSr has a unique F,-critical point, and that the surfaces are convex. The same results were obtained independently by Stein [1]; for n = 2 they were established already in Lavrent'evLyusternik El]. Ffiry and Rêdei [1] established also that
MEASURES OF SYMMETRY FOR CONVEX SETS
F.(4') = F:(4') = (n
=
+ ir E
(_1)'('t
+
255 1—
2ir
r(sintr'dt Jo\ t
/
(The numerical value given for F,(46) is incorrect.) Stein [1] established also
the remarkable fact that the average of F(K), for x ranging over Ke a", is 2*; i.e., From this it follows that F1(K) > 2" for
KeJr.
Special investigations were made on the lower bound of F.(K) for plane sets K of constant width. Besicovitch [2] showed that each such set K has measure of symmetry F6(K) 0.840.... This result was extended by Eggleston [1].
The following examply, due to W. J. Firey, shows that the KovnerBesicovitch measure of symmetry does not satisfy the superminimality condition (*) (see p. 243), and not even the weaker condition (***) (p. 244). Let KA
be the ."truncated" square (see Figure 14); then, for A
1/2, it
is easy to
1
FIGURE 14
show that F.(KA) =
(2 — 2A)/(2 — At)
disc, Fe(KA + öBt) will be less than
On the other hand, if B6 is the unit provided ö is small enough and
A 21(1 + n') for Ke .R".
BRANKO GRUNBAUM
(ii) The derived measure F based on the hexagonal points of Ke.Rt. This derived measure was the first variant of F6 to be studied. Both Besicovitch [1] and Fâry [1] proved F(K) 2/3 for each hexagonal point h of Ke and in this way established F6(K) 2/3. The same estimate of F(K)
was obtained also by Yagloin-Boltyanskii [1] and Linis [1]. The result, as established by these authors, is even stronger: Every affine-regular hexagon inscribed in has an area of at least 2/3 the area of K. (iii) The derived measure F6' based on sixpartite points was considered by Eggleston ([2], reproduced in [3; 5J); he proved F(K) 2/3 for every sixpartite point s of (iv) The derived, measure F based on the center of the circumellipsoid seems to be the simplest example of a geometrically "plausible" measure of symmetry for which, even in E2, the simplices (= triangles) are not the sets with minimal measure of symmetry. Though the lower bound of for Ke seems to be unknown, the set K of Figure 15 can be shown to satisfy =
+ 2ir
FIGURE 15
6.6.
For any convex body Ke SQ" and be the minimal convex body with center p. containing K. We define F71'(K) = V(K)/ V(K[P]) and F7(K) = max {Ff(K): P E K). The lower bound F7(K) 1/2 for K€ seems to have been established first by Estermann [2]; he also proved that triangles are the only extremal sets. These results were obtained also by Levi [1], and the bound 1/2 also by Fáry [1] and Yaglom-Boltyanskii [1J. Estermann [2] asked whether F7(K) 1/3 ESTERMANN'S MEASURE OF SYMMETRY.
any point p e K let
MEASURES OF SYMMETRY FOR CONVEX SETS
257
The estimate F7(K) 1/6 for follows from a result on circumscribed parallelotopes (see § 2). for K e Sr was established by Rogers and Shephard The bound F7(K) > [1; 3]. As pointed out by Ffiry-Rêdei [1] (and also Rogers-Shephard [3]) the F7-critical set of K€ Sr may be n-dimensional (e.g., if K is a simplex); they also found some regularity conditions on K which guarantee that the critical set of K consists of one point only. For the n-dimensional simplex Fáry-Rédei [1] determined
the problem is still open.
for
F7(S*)
= Levi The only derived measure of F7 which has been considered is for KG Sr. [1] proved that F/'(K)> For some extremal problems related to Estermann's measure of symmetry
see Rogers-Shephard [2]. 6.7.
SOME ANALOGUES OF THE KOYNER-BESICOVITCH AND ESTERMANN MEASURES OF
and be defined similarly For a convex body KG let to F6(K) and F7(K), except that the perimeters of K and K(J.') [resp. KEp)J areas). Due to the additivity of perimeterare compared (instead of length, both F and F satisfy the superminimality condition. Therefore it is for triangles, in order enough to determine the lower bounds of F' and It is trivial that F.4'(K) 2/3, with to have the lower bounds for all K€ SYMMETRY.
2/3 only if K is an equilateral triangle. Probably (v. ah equality only if K is an equilateral triangle), but the inequality does
F7 (K) = r.
seem to have been proved yet.1' The
sets may be 2-dimensional,
'd they (and also the level-lines in general) are convex (see Figure 16). It may be remarked that a straightforward application of Bieberbach's [1] isoperimetric theorem yields F6'(K) > 2/sr for Ke Se'. The fact that each of
K
FIGURE 16
" See Note 6 on p. 264.
BRANKO GRONBAUM
258
the conditions F."'(K) 1 or F7(K) = 1 implies that K has a center follows at once from the well-khown strict monotonicity of perimeter-length of convex sets.
The measure F." was proposed (oral communication) by W. J. Firey; he also posed the analogous questions for higher dimensions. We did not reproduce them here since no results about them seem to be known. 6.8. THE DIFFERENCE-BODY MEASURE OF SYMMETRY. For a convex body Ke R' let K* = (1/2)[K + (—K)]. (The set
is usually called the difference-body,
or the vector-body, of K.) Then V(K*) V(K), with equality if and only if K is centrally symmetric. It follows that F.(K) = V(K)/ V(K) is an affine'invariant measure of symmetry. Blaschke [4] raised the problem of determining the lower bounds of F,(K) for Ke W'. For n = 2 the relation F.(K) Rademacher [1] and Estermann (1; 2]. The case n — 3 2/3 was established was settled by Estermann (2] and SUss [1]. Partial results on the general problem were obtained by Bonnesen-Fenchel [1, p. 105] and Godbersen [1]. The complete solution was found by Rogers and Shephard [1], which proved that F,(K)
for Ke 2".
The bounds are attained for simplices only. Modifications of their proof are given in Rogers-Shephard (3] and Eggleston [5]. Groemer [2] proved that F.(K) 2''/(2" — 1) whenever Ke SI' is a pyramid; equality holds if the base is centrally symmetric; for n = 3 this result is due to Estermann [21. 69. THE ROGERS-SHEPHARD MEASURE OF SYMMETRY. Let K be a convex body orthogoin E'; let B' be imbedded in E"1 and let e be a unit vector in nal to E". Rogers and Shephard (3] defined the "associated" (n + 1)-dimensional convex body K* as the convex hull of K U {e + (—K)). Then F,(K) = is an affine-invariant measure of symmetry. Rogers and Shephard 131 established F.(K) (n + 1)2" for Ke 2", with equality if K is
a simplex.
Some extremal problems which possibly lead to measures of symmetry. The distance-function d on the space E of affinely equivalent classes of convex bodies in B" (see § 2) leads to many problems, some of which are relevant in functional analysis. The following results on d2 are known (4 denotes a simplex, C hypercube, Q hyper-octahedron, B ball, M any centrally symmetric convex body, K any convex body; superscripts denote dimensions; H is the regular hexagon): (1) d,(B', K') d2(B', 4") = In d2(B", M') C') = d2(B', Q') = 1/2 ln n, John [11, Leichtweiss [1], (2) d2(C', M') Inn; M') In n , Day [1], Lenz (1], Taylor [1], (3) d2(Ct, M2) H) = in 3/2; d2(H, M') in 3/2, Asplund [1], 7.
1.
(4)
K2)
d8(C2, 4) = in 2,
MEASURES OF SYMMETRY FOR CONVEX SETS (5)
d2(if, K')
M') =
In n.
To prove (4), it is sufficient to verify the easy equality d2(C, I) = in 2 and, for K * 42, to consider the quadrangle A of maximal area contained in K;
then the parallels to the diagonals of A through the vertices of A are
supporting lines of K; they form a parallelogram T(C') such that a translate of T(C)/2 is contained in A C K. In order to prove the inequality in (5) one has only to consider a simplex 4' of maximal volume contained in K; a translate of —nd" then contains K. To prove that d,(4', M") = in n, we remark first that if M" contains some T(4") then it contains also a translate of — T(4"); but the smallest I A I for
which a translate of AT(f) contains both T(4') and a translate of — T(4") is IAI = n. Some of the interesting open problems are: Is the d,-diameter of 51 equal in n? Is max d,(C', K') = In n? What is the d,-diameter of the space (consisting of all centrally symmetric members of .Q)? Is max d,(M", C') = in 2.) in (n + 1)/2? (It can be shown that d,(C', Q3)
The relation of the distance-function d, to a measure of symmetry arises from the following conjecture, which is open even for n = 2: d,(4', K') = In n if and only if K" has a center of symmetry.1 If this conjecture is true, then f(K) = d,(K4')/ln n is a measure of symmetry derived in a particularly simple way from an aft me-invariant distance-function (see § 5 for a more complicated relationship of a similar kind).
FIGURE 17
A great variety of problems may be posed about functionais similar to d,. We mention the following two: (a) Define K,) in the same way as d,(K1, K,) was defined, but with the additional assumption that the centroids of K, and T(AK1) coincide. The 42) is are not known even for E'. The value of seen from Figure 17 to be not greater than in 5/2; it can be shown that actually
bounds on 1
See
Note 8 on p. 264.
BRANKO GRUNBAUM
This is probably the least upper bound for d(K2, 42), and possibly also for d'(KL K). (b) Consider the following procedure, originating from a problem on projections in Minkowski spaces. (We note that it is easy to use a slight modification of the following definition in order to obtain stilL another distance for the space of affine classes of convex bodies in E*; but the corresponding problems seem to be extremely unmanageable.) For a convex body K = — K c E* let P(K) be the greatest lower bound of positive reals 2 for which L a1 = 1, such there exist affine transformations T,, and reals 0 C AK. that K c and It is known (Grünbaum [2], Rutovitz Li)) that d*(Cz, 42) = in 5/2.
= n.21h1(
= 1)/21)
and
PIB*\
Determine I',,
max P(K) for K =
—K C E*.
In particular, is P2 = 4/3, results on d3 that P 3 there seems to be even no conjecture about the value of E*(n). 7. Finite setB. A completely different approach, also of an elementary nature, was used in order to establish an affirmative solution to Borsuk's problem for finite sets in E2 and E (but the problem is still open for finite sets in E", n > 3). with diameter A = 1, Considering a set A, consisting of k points in
one may ask how many pairs of points of A are at distance 1. Let us denote
the maximal possible number of pairs by F(k, n). A very simple proof of F(k, 2) = k was given by Erdös [I), where also the question about F(k, 3) is raised and a conjecture of A. Vazsonyi that F(k, 3) = 2k — 2 is quoted. In the
wake of Erdös' travels in 1956, three independent proofs of Vázsonyi's conjecture were published (Grtinbaum [1], Heppes [1], Straszewicz [1]). From the values F(k, 2) = k [resp. F(k, 3) = 2k — 2] it is easy to derive the a direct affirmative answer to Borsuk's problem for finite sets in E2 and proof for the three-dimensional case, along similar lines, was given by HeppesRêvész [1).
In higher dimensions the values of F(k, n) are not known (Hadwiger [8)).
For every k> n
2 it is easy to construct in E* sets which show that
F(k, n) (n — 1)k — (n + lXn — 2)/2 (Hadwiger (8, Appendix)).8 Even if the conjecture that F(k, n) equals to this expression were established, it probably would not lead to an answer to Borsuk's problem; this approach seems feasible only for those n for which F(k, n) k(n + 1)12, i.e., n 3.
The problem of determining F(k, n) for n > 3 seems to be very difficult, 8
See Note 3 on p. 284. See Note 4 on p. 284.
BORSUK'S PROBLEM AND RELATED QUESTIONS
mainly because of the essential difference in the strucfure of the graphs determined by the edges of convex polyhedra in E' for n 3 and n > 3 (noticed already in Carathéodory [1]; for a review of the results known about such 'polyhedral graphs' see Griinbaum-Motzkin (1]). 8. Other related problemB. The theorem of Borsuk-Lusternik-Schnirelmann may be formulated as follows: If S" is covered by the union of n + 1 closed sets, at least one of the sets contains a pair of points the spherical distance between which is it. This result may be considered as a special case of the following problem (in which the number of alternatives could be increased easily).
What is the greatest integer m such that for every covering of (1)
sets
by m (2) closed sets (3)
M
closed congruent sets
at least one of the sets M. contains, for (1)
a given
every fixed9 (3) all (2)
positive 0 not exceeding the diameter of the space, a pair of points at distance 0.
To give an example of the rather self-explanatory notation we shall use, the theorem of Borsuk-Lusternik-Schnirelmann could be formulated, with 0 = it, 2, 1) n + 1; the simplicial decomposition of shows that equality as
holds in this relation. The conjecture m(S*, 2,3) = n + 1 (i.e., in every covering of S" by n + 1 closed sets, the points of at least one of the sets realize all distances, and the number n + 1 is maximal with respect to that property) has been forwarded by Hadwiger [2; 10], and established by Hadwiger [1] for n = 1,2. The weaker proposition 2,2) = n + 1 was proved (in a more general setting) by H. Hopf [1]. Hadwiger [31 was able to show that 2, 2) 3n — 1, where the asterisk * indicates that only distances o with cos 0 —1/n are considered. Under the same additional condition on 0, Hadwiger [2] showed 2,3) = n + 1. that For Euclidean spaces, Hadwiger [2] proved that 2,3) n + 1. Except
for n = 1, it is not known whether n + 1 is the best bound; the best upper estimate for n = 2 seems to be m(E2, 2, 3) m(E*, 3,3) 6 (Hadwiger-Debrunner [2]).
A related result due to Hadwiger [3] is
n
2.
For a discussion of other variants in case of dimensions one and two see Hadwiger-Debrunner [2].
Another unsolved problem is the determination of question whether .' Mc depenthng on ö.
2, 2) =
2, 3) seems to be open.
2,2); even the
280
BRANKOGRUNBAUM
Problems like that of m(E*, 1, 1) concerning coverings by (or decompositions
into) arbitrary sets may be rephrased conveniently in terms of graphs and their colorings. Let a 0-graph be any graph whose nodes are (some or all) points of ER (or Sn), two nodes being connected by an edge if and only if 1, 1) their distance is 0. A question equivalent to the determination of is to find c,, = 1 + m(E*, 1, 1), the maximum of the color-numbers of 0-graphs in ER (The color-number of a graph is the least integer k such that the nodes of the graph may be divided into k classes, with no class containing two nodes connected by an edge.) By a theorem of de Bruijn-Erdös [1], it is sufficient to determine the color-numbers of finite 0-graphs. A simple example (Moser-Moser [1J) shows that c2 4 (see Figure 3). Similar examples in higher dimensions show that n + 2. The exact value of c,, is not known even for n = 2; the best upper bound known in this case is c2 7 by the result of
FIGURE 3
Hadwiger-Debrunner t21 mentioned before (since 1 + m(E2, 1, 1) m(E2, 3, 3) + 1 7). For additional comments on c,, see Klee [1), where the
question about the value of c, is attributed to E. Nelson. The numbers c defined for S" in analogy to the definition of depend = 2, while on 0; thus obviously fl + 2 if C,, satisfies cos 0,, = —1/n. It seems even not to be known whether C,, = l.u.b. {c(O): 0 2. We believe it is possible but the problem requires much more detailed analysis than is required for the planar case. In order to make this account comparatively self-contained, we will review, with geometrical emphasis, the theory of diametral lines which led us to define the outwardly simple line families. We also present certain relevant 291
292
P. C. HAMMER
results proved in Planar line families. I and Planar line families. II 122; 23], and we use other results proved by T. J. Smith in his thesis (1961). It is necessary to summarize our work on associated convex bodies, AB [211, and we alter the form here. While we stick to the central theme in general, we could not bring ourselves to omit the characterizations of classes of equivalent breadth curves by arc length functions. The finite construction of relatively constant breadth curves geometrically could have been omitted. However, these constructions illuminate the theory and permit constructions in elementary geometry. The details of carrying out the limiting process are omitted. II. Notations and conventions. Except as noted we are concerned here with the real affine plane as a vector space in which we assume a Cartesian coordinate system. Lower case letters, a, b, c, d, v, x, y, z, are used to designate
points (vectors) or vector functions. The letter u is specifically reserved to represent points on the unit circle usually in form u(O) = (cos 0, sin 0). Then u'(O) = u(0 + t/2) = (— sin 0, cos 0), and the dot product u(6) . u'(O) = 0. Real numbers or real-valued functions are designated by f, g, h, p, q, i-, s, I. Curves representable in the usual polar coordinate system are given by p(0)u(6) where p(O) is a real-valued function.
Sets except lines, line segments, and curves, are indicated by capitals X, Y, Z, while convex bodies are designated by A, B, C, D and their boundary
curves are indicated by the corresponding Greek letters a,
T,ö.
Scalar
multiplication and direct sum of sets are defined by
IX= {tx:xeX,t a real number), X+ Y= Y}, We note that t0X0 + t,X1 is a convex set when X0 and K1 are convex sets.
We indicate a line parallel to the vector u(0) by m(0). Such a line is uniquely determined by a real number such that P(0)u'(8) is the foot of the perpendicular from the origin 0 to m(6). If p(0) is a function of 0 defined on (—co, 00) then determines a family of lines and p is called the pedal function of F(p). If p has a derivative at each value 0 then the set of points v(8) p(8)u'(O) — P'(8)u(0) is called the envelope of F(p). A S is the closed set bounded by two parallel lines or, in degenerate cases, a strip may be a line. If K is a bounded set then in each direction u there exists a minimal closed strip S(u) containing X with sides orthogonal to u. This is called a supporting strip of X. The Euclidean breadth of X in the direction u is the distance between the parallel sides of S(u). Two bounded sets X and Y are of equivalent breadth provided each supporting strip of one is a translate of a supporting strip of the other. Note that X and Y are of equivalent breadth if and only if their closed convex hulls are of equivalent breadth. Observe that equivalence of breadth is an affine invariant independent of a definition of breadth. III. Diametral chords and central symmetrization. Let C be a convex
CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH
293
body with boundary curve r. In each complete family of parallel chords of C there is at least one of maximal (relative) length. Such a maximal length chord is called a diametral chord of C. Equivalently any chord of C which crosses a supporting strip of C is a diametral chord. 3.1 THEOREM (H.1IIMER AB). The diametral chords of a convex body C cover the body. PROOF. Since every boundary point of C lies on a supporting line of C it is clear that the diametral chords of C cover r. If x is an interior point of C let r be the maximum positive number such that (1 + r)x — rC C and let b be a common boundary point of C and (1 + r)x — rC. Then define a by (1 + r)x — ra = b. It follows that ab is a diametral chord of C through x since if m is a supporting line of C through b then the line m1 such that (1 + r)x — rm1 = m is a supporting line of C through a and parallel to m. Hence C is covered by its diametral chords. Q.E.D. REMARKS. This proof holds for n-space. Note that (1 + r)x — C is the reflection of C through x. DEFINITION. The set = (X — X)/2 is called the symmetroid of the set X and the transformation is called central or pdint symmetrization. 3.2 THEOREM. Let C be •a convex body and let C be its symmetroid. Let r and r5 be the respective boundary curves. The following state,nents hold.
(1) C and
C
breadth
as a unit circle. (2) Let ab be a diametral chord of C crossing a supporting strip S of C. Then [(a — b)/2, (b — a)/2] is a diamefral chord of and S — (a + b)12 is a supporting strip of C5. Hence parallel diametral chords of C and C* have the Minkowsk ian metric based on
the same length. (3) Let D be a convex body with symmetroid D5. Then C and D are equivalent in breadth if and only if C' = D'. Moreover C and D are equivalent in
breadth if and only if their Parallel diametral chords have the same length always. PRooF. (1) and (2). If S is a supporting strip of C crossed by a diametral chord ab then S' since S C. But S5 is a translate, say, S — (a + b)/2 of S and since (a — b)/2, (a + b)12 C5 and the chord joining these points cross S5. we have that S is a supporting strip of C5. Hence C and C5
are equivalent in breadth and have parallel diametral chords of equal length. (3) This result is a corollary of (1) and (2) since D is equivalent in breadth to D5 and hence to C5 if C and D are equivalent in breadth. But two symmetrical convex bodies with the same center point are obviously equivalent in breadth if and only if they are identical. Q.E.D. REMARKS. That C and C * are equivalent in breadth was certainly known to Meissner and quite likely to Minkowski. The use of diametral chords we
have not seen outside our work, although in the theory they play the role comparable to the binormals of constant breadth curves. That parallel
P. C. HAMMER
294
diametral chords go with parallel supporting lines for equivalent breadth curves
is a feature of the differential equivalence we exploit later. be a convex curve sym,netric with respect to 0 and let Let 3.3 B be the convex body P determines. Then a necessary and sufficient condition that a convex body C be of constant breadth in the Minkowskian metric with as unit circle is thaf there be a Positive number r such that C * = rB. Then
length of chord of the
PROOF.
a
diametral chord of C is r times that of a parallel diametral
If the breadth of C relative to
is 2r > 0, then the symmetroid of breadth 2r relative to and hence rB = C*.
C
Q.E.D. IV. Diametral lines and associated convex bodies. A line containing a
diametral choed of a convex body C is called a diametral line of C. 4.1 THEOREM (HAMMER AB). Let C be a convex body gnd let F be the family of diametral lines of C. The following statements hold. (1) F covers the extended plane. (2) F simply covers the exterior of C, including infinite points, if and only if r contains no pair of parallel opposite line segments. COMMENTS. We state this theorem since it shows that we need to select, in certain cases, a subfamily of the diametral line family. A family F0 of diametral lines of a convex body C is said to represent C provided it simply covers the exterior of C, including infinite points. If r contains no parallel
opposite line segments, then F0 must contain all diametral lines of C. Otherwise, there is an infinity of families of diametral lines representing C. For some purposes we want a unique family of diametral lines. The best choice
from several points of view results when we choose F0 as near a pencil as possible. That is, suppose ab and cd are parallel opposite line segments in r so that a — b and c — d have the same sense. Then ad and be are diametral chords of C which intersect in a point x interior to C. We choose those diametral lines of C which pass through x to represent those which cross between ab and cd. With this choice always made, F0 is called the essential diametral line family of C. 4.2 THEOREM.
Let p be the pedal function of a diametral line family F0
which represents the convex body C. such that r is represented by x(O)
Then there exists a unique function f(O)
p(8)u'(O) + f(O)u(8)
where f(O + iv) + f(O) > 0. The function f satisfies f(O) + p'(O) for which p has a derivative.
0 for all 6
Paoor. Since the diametral line family F0 covers the extended plane and since on each line in F there are two end points of a diametral chord, we
have chosen f(O) so that x(8) is the point on m(6) making
CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH
= [f(0) + f(8 + ir)]u(0) have the same sense as u(8). Now, C contains the intersection points of line pairs from F0 by choice of F0. However, as a theorem [Theorem 5.21 esx(8) — x(0 + it)
tablished by T. J. Smith shows, the closed convex hull D of the set of intersection points for F0 is the same as the closed convex hull of the set (p(0)u'(6) — P'(0)u(8)}
for all 0 for which p has a derivative. Hence f(0)
—
p'(0) necessarily since
Q.E.D. No'rE. As the referee kindly pointed out, we should mention that the function f(0) in the representation (1) is necessarily Lipschitzian and, since p and I are periodic and Lipschitzian, they are uniformly Lipschitzian. 4.3 THEOREM. Let r be a convex curve represented by x(0) as in (1) Theorem 4.2. Then for almost all (2) I x'(O + it) I x'(O) + I x'(8) x'(O + it) = 0 PROOF. Since r is convex the difference quotient vector (x(0 + 40) — x(0))/40 will approach a limiting direction (or be 0) as 48 approaches zero through positive (or negative) values. However, the derivative x'(O) may not exist also because of the fact that a unique limiting length may not exist. Now, as proved in Planar line families. II, p is a Lipschitzian function. Hence p has a derivative almost everywhere. Then if we exclude from consideration those values of 0 where P has no derivative and also those values of 0, 8 + ,v where the left and right difference quotients of x(8) do not agree in limiting direction, we have excluded at most a set of measure zero. At the remaining values of 0 we have that x'(8) exists. If x'(O) = 0 or x'(O + it) = 0, (2) holds by default. Otherwise, neither x'(8) = 0 nor x'(O + iv) = 0. Then x'(O) and x'(O + iv) are para:lel and in the opposite sense since x(0), x(6 + iv) are I
end points of a diametral chord and hence (2) holds almost everywhere. Q.E.D. 4.4
THEOREM.
Let
be
the boundary of the symmetroid C* of a convex body be represented
Let F0 be a diametral line family representing C and let by (1) as in Theorem 4.2. Then is represented by C.
y(8)
= x'(O) =
+ f(8 + ir)Ju(8)
y'(8)
almost everywhere.
REMARKS. The above theorem is a corollary of Theorem 3.1 and the representation Theorem 4.2. The equation (4) expresses the fact that for corresponding values of 8, and have usually parallel supporting lines (or
296
P. C. HAMMER
else x'(e) = 0). Note that I y'(8) I is bounded from zero and y'(O) exists except
for possibly a countable set of values of 0. DEFINITION. Let C = C(1) be a convex body. Let C(r) be the set obtained from C by extending each of its diametral chords about its midpoint by a constant ratio r> 1. Then C(r) is called an associated convex body of C and of C(r) is called a parallel curve of r r(l). the boundary curve It is clear that the supporting strips of C(r) are obtained by expanding the supporting strips of C about their midlines by the ratio r. Hence C(r) is a convex body and C(r) is of constant breadth (=2r) relative to T*(1) as unit circle. It is of interest to state the relationship of the boundary of rfr) to r = r(l). 4.5 THEOREM. Let C(1) be a convex body and let C(r) be an associated convex body of C(1) for some r> 1. The following statements hold. (1) The curves i(r) and r(l) are a distance r — 1 apart relative to T*(1) as unit circle. Hence C(r) = C(l) + ( — 1)C*(1). (2) The set ((r — 1)/2)C*(1) may be translated around the closed ring between r(l) and r(r) touching both curves always. Hence this set is the maximal symmetric set with this property. Hence also 1 1+r V C*(1) : xe C(r) = C(1) U {x +
)}
(3) (4)
A diametral line of C(1) is a diametral line of C(r).
Lim,-... (1/r)Cfr) = C*(1). REMARKS. The foregoing statements are readily verified and may be directly
extended to n-space. Now we consider internal associated convex bodies of C = C(1). By reducing each diametral chord of C (and each supporting strip)
about its midpoint (midline) with a ratio r < 1, we obtain a convex body D(r) as the intersection of the derived supporting strips. But, in general, the symmetroid D*(r) c rC*(1). There exists a unique minimal number r0 0 such that D(ro) = roC*(1). Then we define C(r) = D(r) for r0 r 1, and
we call these bodies the internal associated convex bodies of C. Then the body C(r,) is called the irreducible subbody of C(1) unless r0 = 0, in which case C(1) is symmetrical with respect to a point [= C(r0)J. Letting A = C(ro) if r0 > 0 we have that the external associated bodies A(r) of A for r> 1 are given by A(r) = C(rro) and A*(r) = rroC*(1). The diametral lines of A(1) are diametral lines of A(r) for all r> 1. In Hammer AB it is shown that the essential diametral line family of A(r) is the same as that of A(1) = A. To generate equivalent breadth convex bodies from others, the following theorem is stated. 4.6 THEoREM. If A is equivalent in breadth to B and C is equivalent in breadth to D, then s0A + s1C is equivalent in breadth to s0B + s1D.
Let A and C be of constant breadth relative to a symmetrical convex set Then s0A + s1C is of constant breadth relative to B for all real s0, s1. V. Outwardly simple line families. A family of lines which simply covers
B.
CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH
the exterior of some bounded region, including infinite points, is called outwardly simple or a quasipencil. In higher dimensions the concepts must include continuity and the quasipencil is a more restricted type of line family than the outwardly simple type. In the plane continuity is implied in the definition and quasipencils coincide with outwardly simple line families. Hammer and Sobczyk in Planar line families. I and II give geometric and analytic characterizations of outwardly simple line families. We restate a few of these results here in slightly modified form. Pictorially, an outwardly simple (o.s.) line family is generated as follows.
Two runners x and y run on a circular track according to the following They start at the same time at diametrically opposite points and run continuously without stopping in the same sense and at the end of one hour x is to be at the starting position of y and y is to be at the starting position of x. If x and y carry a line between them then an outwardly simple line line. Conversely every outwardly simple line family is generated by family may be so obtained. This picture may be extended to a closed convex curve with stops of runners permitted. rules.
5.1
Let F be a family of lines with pedal function p.
Then
necessary and sufficient conditions that F be outwardly simple are that (1)
+ it) = —p(O)
for all 6, and
(2) p be uniformly Lipschitzian.
REMARKS. The first condition assures that infinite points (directions) are simply covered, whereas the second gives both a continuity condition and boundedness of the set of intersection points of line pairs from F. Note that the derivative p'(O) exists almost everywhere and is uniformly bounded. DEFINITION. The set of all points v(O) = P(O)u'(O) —
for which p has a derivative p'(O) is called the virtual envelope of F. The
following theorem proved in detail by T. J. Smith in his thesis is useful later. 5.2 THEOREM (SMITH). Let F be an o.s. line family. Let X be the set of all intersection points of line pairs from F. Then the closed convex hull of X is identical wi/h the closed convex hull of the virtual envelope of F. The set of pedal functions for all o.s. line families form a linear space which is complete
but inseparable with the norm
liP
II
=
sup v'(P(O)P
+ [p'(O)J'.
VI. Analytic representation of curves of constant Mlnkowskl breadth. 6.1 Let p(O)u(O) be the symmetrical convex unit circle for a Minkowski metric. Let p be the pedal function of an outwardly simple line
family F. Then there is a constant number q such that the curve r = r(q) given by
P. C. HAMMER
298
x(O) = p(O)u'(O) +
PP
+
with F as a representing is a convex curve of constant breadth relative to family of diametral lines. Furthermore every convex curve of constant breadth relative to may be so represented (by proPer choice of F and q). PRoOF. First we show that if F is a family of diametral lines representing a convex curve T of constant breadth relative to then r is representable in form (1).
Let r be represented in form
x(8) = p(O)u'(O) + f(8)p(O)u(t9)
where f(O) + f(8 +
a positive constant. Now for almost all 0 there is
is
a nonnegative number g(0) such that x'(O) = g(0)[p(0)u(0)]'.
Substituting from (2) in (3) and omitting the arguments, we find, on setting coefficients of u'(O) and u(0) equal to zero, + fp = gp, —P +f'p +fp' = gp'.
Eliminating g and solving for f' we have
Pt'
f1
P
Hence for some constant q t.e
"t'
Jo
P
P
But from (4) it follows that necessarily
4+ Hence
P(°)
any convex curve
of constant breadth relative to p is representable
by (1).
Now, however, suppose F is a given outwardly simple line family with be the minimum real number q such that
pedal function p. Let qo
p'(0) p(0)
for all 0.
Now let q >
o
P
and consider the curve r represented by (1). Then
is a simple closed curve containing the virtual envelope of F since fp> —p' and hence r is simply covered by F (Smith's theorem). Moreover x(0) — x(0 + 2r)
"I' dcti]p(e)
= + and hence the length of chords of r on F is Constant relative to 46. Moreover
r is convex since there is g(6) > 0 and
x'(O) = g(0)[p(0)u(0)J'
CONVEX CURVES OF CONSTANT MINKOWSKI BREADTH
299
for almost all 0 and hence r is locally convex and hence convex. The value of q may now range down to q0 and to + to obtain all convex curves represented by F and of constant breadth relative to P. In the case that F are symmetrical and r(qo) reduces to a single point. is a pencil the curves Q.E.D. NOTES.
Letting p(0)
1 we obtain all constant breadth curves in the form
we previously gave: x(0)
=
+ (q +
0
In the above proof we used the facts that
+
= —p(0) and p(0 + ir) =
The generality of representation was implemented by direct definition of the outwardly simple line families and by their representation using pedal functions. Other attempts to obtain the constant breadth curves by starting with the envelope curve failed to achieve generality since the specifications for this curve were too stringent to allow all outwardly simple line families. It is conceivable that the representation of Meissner could, in view of later development of theory of Fourier series, be shown to represent the curves of relatively constant breadth. If we write p(O).
=
cos (2j —
1)0
+ Eb, sin (2] —
1)0
and if these series are chosen so that P(0) is Lipschitzian one may obtain many outwardly simple line families. However, for applications to kinematics it may be that using finite Fourier series or other representations of p will be adequate. It will be observed that (1) will still give closed curves intimately related to a symmetrical curve P which is not necessarily convex. Since the transformation from a convex curve 5 to a symmetroid curve is theoretically simple we are in a position to obtain curves r equivalent in breadth to a given curve 5. In doing this, of course, the outwardly simple line family F cannot be chosen arbitrarily. VII. Arc length function of equivalent breadth curves. Let C be a convex body and let F be an o.s. family of diametral lines of C which represents C. Define the function s(0) as the sum of lengths of the (two) arcs of r between rn(O) and m(0). Then s(0) is called the arc length function of r. Note that s(,t) is the circumference of C. 7.1 THEOREM. Two convex bodies C and D are equivalent in breadth if and only if they have identical arc length functions. have the same arc length PROOF. We show that C and its symmetroid function regardless of the representing family F used. With p as the pedal function of F we represent r by x(0) = P(0)u'(0) + f(0)p(0)u(0)
where p(O)u(O)
is the vector function of
and hence f(0) +f(0 + ir) =
2.
P. C. HAMMER
300
Now s(8)
=
II x'(8)
I
Jo
÷ I x'(O + it) j}dO = I I x'(8) — x'(O + it) I dO Jo
since x'(O) and x'(O + it) are parallel the derivative vectors exist. But I x'(O) — x'(O + it)
I
in opposite sense unless one is zero when
= Ix(0)
—
x(8 + it))' I =
2 I [p(8)u(0)J' I
since f(0) + 1(8 + it) = 2. Hence s(O) = 2S1 [p(O)u(O)]' do which is the arc for the pencil of diametral lines through 0. Hence if length function of C and D are of equivalent breadth they have identical arc length functions. and Now suppose C and D have the same arc length function s(8). Let 5* be the boundaries of the respective symmetroids of C and D. Then since
have the same circumference and are centered on 0 they must and intersect in at least four points, two of which make a central angle with 0 5* there must be a pair of points (say at 01 < 0,, less than it. Hence if say, is
(4.17)
0
= (li
Yii
Y12 + Y22 = d2 — d1 Yia
+ )izs
+
Yin
+
=
+ynn
dn
Now consider the following problem: (4.18)
+ xi,,) +
p1(x11 + ...
where the variables x
,
-, x,,,,,
-
Minimize
+
y,
d2,
d3 —
-,
-. + v,,,,
X2n)
+
- -
-+
satisfy (4.8), (4.17) and
ON SIMPLE LINEAR PROGRAMMING PROBLEMS
325
(4.19)
+
+X2,,
+
+Yzn xe,,, +
=Oz,
y,.,, =
It can be shown that: satisfying (4.8), (4.17) and (4.19), the (i) given any variables x,, and variables x obtained from (4.7), (4.9) and (4.6) satisfy (4.2) and (4.4) and yield a value for (4.1) identical with the value of (4.18); (ii) conversely, given any variables x, satisfying (4.2) and (4.4), one can find variables x,5 and Yu satisfying (4.8), (4.17) and (4.19), such that (4.1) equals (4.18); (iii) the conditions (4.8), (4.17) and (4.19) are those of a transportation problem which can be solved by inspection because an appropriate Monge sequence can be identified. The proof of (iii) will occupy the next section. The proof of (ii) is somewhat long (see [3]) and will be omitted. The proof of (i) will now be given.
Observe that the content of (i) and (ii) jointly is that our transformed problem is equivalent to the original one. PROOF OF (i). It is clear that, using (4.7), (4.9) and (4.6), one obtains (4.4) and the equality of (4.1) and (4.18) immediately. What remains is (4.2). To prove the left side of (4.2) observe that-
=
+
(Xis
+ x22) + - - + -
+ -.
+
Xtt)
A similar discussion shows that
and ((4.11) and (4.13)) this implies (4.10) and hence the right side of (4.2).
This completes our construction of the transformed problem. Note that this construction required not only the notion of tagging production in any given period with the period whose requirements it would help satisfy, but also the notion of tagging unused capacity in any period with some period whose requirements it would not help satisfy.
The usefulness of this idea in the present problem will be apparent in the
seems such a strange thought that there may very well be sequel, but other opportunites for using it when its meaning has been absorbed. 5. ApplicatIon of the Monge sequence. To fix our ideas, consider the case n = 4. All the phenomena for general n are already illustrated in this case. Consider the four by eight transportation problem with cost coefficients, row sums and column sums given by the following tableau:
A. J. HOFFMAN
326 (5.1)
b3 — b, d3 —
b4 —
d4 — d3
d1
b2 — b1
d2 — d1
Pt
0
P'
0
Mt M' M6
M Mt
P2
0
0
0
M
0
0
M5
M'
M
0
b1
M3
pt
M2
0
Pt
b3
0
M is an arbitrarily large positive number. Notice first that this transportation problem has non-negative row and column sums, and satisfies the condition that the sum of the row sums equals the sum of the column sums, for the sum of the column sums is b4 + d4. By (4.3) and (4.14), this is
b4+aj+a2+a3+a4—b4=a1+-•.+a1, which is the sum of the row sums. The odd columns refer to variables the even columns refer to variables The large coefficients Mk compel certain variables to be zero. It is clear that this transportation problem is then identical with (4.18).
Next, arrange the 32 elements of the cost matrix in a sequence by the following rule: (i) list first the elements of the first column in ascending order of magnitude, (ii) list the elements of the second column in ascending order of magnitude, (iii) list the elements the third column in ascending order of magnitude, (iv) list the elements of the fourth column by first putting the zeroes with indices whose corresponding P's are in descending order of magnitude, then the powers of M in ascending order, (v) list the elements of the fifth column in ascending order of magnitude, (vi)
list the elements of the sixth column by first putting the zeroes with
indices whose corresponding p's are in descending order of magnitude, then M, (vii) list the elements of the seventh column in ascending order of magnitude, (viii) list the elements of the eighth column in any order. Then one sees
that the stipulations (2.2) have been satisfied for this sequence, so the algorithm of § 2 applies. It can he shown, of course, that if inequalities (4.2) and (4.4) are consistent, the algorithm will never choose a positive if is a power of M. 6. Remark8. The first transformation used above is a special case of a device which appears to have been used for the first time by W. Jacobs in [11]. The second transformation is based on an idea of Prager [14]. (Incidentally, the simple algorithm proposed by Beak [5] for the solution of Prager's
formulation of the caterer problem can be shown to be a special case of the Monge idea; so can the algorithms presented in parts of references Besides these transformations, other tricks are the use of the duality theorem [7] and various devices for standardizing the structure [1]. In the not too distant future, it should be possible to present a catalogue of devices
ON SIMPLE LINEAR PROGRAMMING PROBLEMS
usable in making linear programming problems "simple," whether or not the Monge idea applies; at present, they are too fragmentary to justify listing. By far, the most interesting direction of study is that initiated by Jacobs in [11J. In this instance, he gave an example of how one could minimize on a
set K by minimizing on L K, because it was possible to show that a minimum on L occurred at a point of K. A less ingenious instance of this was given in § 3 above. A comprehensive theory giving classes of cases where such transformations are possible would be very desirable. REFERENCES
1. A. J. Goldman and A. W. Tucker, Theory of linear progra,n,nsng, Linear inequalities and related systems, pp. 53-98, Annals of Mathematics Studies, No. 38, Princeton Univ.
Press, Princeton, N. J., 1956. 2. W. M. Hirsch and A. J. Hoffman, Extreme varieties, concave functions and the fixed charge problem, Comm. Pure Appi. Math. 14 (1961), 355-369. 3. A. J. Hoffman, Some recent appiwations of the theory of linear inequalities to external co,n6inator-ial analysis, Combinatorial analysis, pp. 95-112, Proc. Sympos. Appi. Math., Vol. X, Amer. Math. Soc., Providence, R. I., 1960. 4. G. Monge, Débiai et remblai, Mémoires de l'Académie des Sciences, 1781. 5. E. M. L. Beale, Letter to the editor, Management Sci. 4 (1957), 110. 6. E. M. L. Beale., G. Morton and A. H. Land, Solution of a purchase-storage programme, Operational Research Quarterly 9 (1958), 174-197. 7. A. Charnes and W. W. Cooper, Generalizations of the warehousing model, Operational Research Quarterly 6 (1955), 131-172. 8. C. Derman and M. Klein, Inventory depletion management, Management Sci. 4 (1958), 450-456.
9. M. Fréchet, Sur lee tableaux de correlation dont Lee marges sont donnéee, Ann. Univ. Lyon Sect. A (3) 14 (1951), 53-77. 10. J. W. Gaddum, A. J. Hoffman and D. Sokolowky, On the solution to the caterer problem, Naval Res. Logist. Quart. 1 (1954), 223-229. 11. W. Jacobs, The caterer problem, Naval Res. Logist. Quart. 1 (1954), 154-165. 12. S. M. Johnson, Sequential production planning over time at minimum cost, Management Sd. 3 (1957), 435-437. 13. H. Lighthall, Jr., Sch'duling problems for a multi-commodity production model. Tech. Rep. 2, 1959, Contract Nonr-562(15) for Logistics Branch of 051cc of Naval Research, Brown University, Providence, R. I. 14. W. Prager, On the caterer problem, Management Sci. 3 (1946), 15-23. 15.' W. E. Smith, Various optimizers fir single-stage production, Naval Res. Logist. Quart. 3 (1956), 59-66. INTERNATIONAL BUSINESS MACHINES CORPORATION
TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS BY
SAMUEL KARL!N1 1. Let X and Y be real intervals and let K(x, y) be a bounded measurable function defined on the rectangle X ® Y. It is useful and interesting to find conditions on K which imply that the transformation
xeX,
9— Tf
sends bounded convex functions into convex functions. Our objective is to establish the relevance and utility of the concept of total positivity and related ordering- properties (see below) to this problem. More precisely, we shall show that the property of total positivity provides simple sufficient conditions
to insure that the transformation T is convexity preserving. We will not enter the question of determining to what extent the converse is true. In this connection, see [9; 31. The
theory of total positive functions and, more generally, sign regular
functions, has been widely applied in several domains of mathematics, statistics and mechanics [1; 2; 3; 13]. An extensive summary of the recent literature
which also contains some extensions of the theorems of this paper can be found in a forthcoming monograph by this author [3]. 1 and 2 we develop alternative definitions and preliminaries dealing In with the property of sign regularity. A number of -theorems are stated in. dicating the relative strength of the various definitions. Basic variation diminishing properties that the transformation (1) inherits when K is suitably sign regular are also stated.
§ 3 is devoted to a discussion of several important
examples which show the usefulness of alternative formulations.
In § 4 generalizations are given and various specific applications are indicated stemming from the original question of the convexity preserving nature of T.
2. Prellminsries. A function K(x, y) of two real variables ranging over sets on the real line X and Y, respectively, is said to be totally positive of order r (abbreviated Ti',) if for all m, 1 m r, and for all' x1 <x, < < ... 0,
n = 0, 1,2, •••. Of course, the series is assumed to converge for all xc X and ye Y. In order to prove that (27) is TP we write it as a Stieltjes integral: (28)
K(x, y)
=
exp [E log u(x)] exp
log v(y)1 dgl'(e)
where c& is a discrete measure with jumps a,, located at n = 0, 1, 2, . As noted above, exp log u(x)J is TP for real and XE X. It remains to apply
(24), thus concluding that K(x, y) is TP, strict when u and v are strictly increasing and ETP(x) when X is an interval, u and ,i(x) > 0. As a concrete illustration, we mention K(x, y) = I.(x y), where denotes the standard Bessel function of imaginary argument. 3. It is frequently simpler to verify that a function K(x, y) is ETP rather
than TP (just as one frequently proves that a function is increasing by proving the stronger fact that it is strongly increasing). As an example, we establish that the important noncentral t-density occurring in statistical theory is ETP an'd a fortiori TP. This density function is of the form A)
=
exp
[—
—
where c is a normalizing constant and a represents a fixed positive parameter. We now prove that p(t, A) is ETP for — oo 0, let v = V'wI2a 1. Then after obvious simplifications p(t, A) =
erA exp (—a(v2/t2)ldci(v)
TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 339
where f and g are strictly positive functions and o(v) is a positive measure whose explicit form we need not write. Applying (24), we obtain that >0
for 1>0 and —oo 0 for all
a x
6. We can prove the folldwing sharper statement
if and only if (5) holds and
i=1,2,".,n—1. This allows certain equalities between the x's and y's. The Green's function K(x, y) of the eigenvalue problem (ry")" = 2y under the boundary conditions
=0 = 0 , ry"
+ a0y
(ry")' + where
cc is
—
=0 =0
,
—
TP. (See [2).) We can prove in this case the
sharper statement that x1,
(35)
•
•,
",YR/
>o
if and only if (5) holds and < Yi+i
(36)
,
0.
We assume that f(x + y) is sign regular (for example, either or RR,) and x) is TP, for t > 0 and x > 0 and obeys the semigroup property that
TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATiONS 341
p(t+s,x)=
(38)
Subject to these conditions we prove that c(t + s) is TI', (RR,) according as f(x + y) is TP, (RR,). To this end, we examine the expression c(i + 5)
(39)
+ s, x)f(x)dx
=
u)f(u + E)dude.
=
The last identity results by applying (38) and then performing obvious manipulations. With the help of the convolution property (24) we obtain that the function ço(s,u)f(u +e)du
sb(s,e)=
is sign regular of the same type as f(u + Now, (39) can be written in the form: c(t + s) =
since o(s, u) was assumed TP,.
e)sl'(s, E)dE.
Another application of the convolution property (24) yields that c(t + s) is appropriately sign regular. The above arguments hold virtually without change in the case that the variable t traverses over the set of positive integers. We summarize this discussion in the statement of the following theorem. THEOREM 6. Let f(x + y) be Ti',. (RR,) for x> 0 and y > 0. Suppose ço(t, s)
is TP,for x>O and t >0 (or t = 1,2,3, defined by (37). discrete case).
and satisfies (38).
Then c(t + s) iS TP, (RRr) for t, s > 0 (t, $ = 1,2,
Let c(t) be
in the
The hypotheses of the above theorem are satisfied in each of the following circumstances:
Let f(x) be a PF, density function of a non-negative random variable. where (a) Let ço(n, be the density function of the sum X1 + + -- + are positive identically and independently distributed random variables whose density function is PF,. - The fact that ç'(n, E) is TP, for n 1 and E > 0 is proved in [7]. The semigroup character of so(n, is clear. (b)
Let
=
1(1) '
Relation (30) and the TI', property may be verified by direct calculation. Let denote the density function of a stable process X(t) with non(C) negative drift whose Laplace transform is exp s > 0 (k = a fixed positive integer rt is shown in (4J that is for t, > 0.
SAMUEL KARLIN
342
Property (38) emerges from the fact that X(t) is a homogeneous process with non-negative drift. The above theorem also enables us to deduce various moment inequalities for PF, densities associated with non-negative random variables. Inueed, let t > 0,
c(t)
where f(E) is a PF density on the positive ray. As observed above in example (b),
ço(i,e)= f'(t)
'
0, satisfies the condition of Theorem 6. Hence c(l + s) is SRR for I, s > inequality of the second order determinant asserts 2a(s + t)
(41)
a(t) + a(2s + I)
0.
s,
,
The I>
0
where a(u) = log c(u) + 1. This shows that a(u) is concave (since a is conIn particular,
tinuous).
a(t) and since a(O) = (42)
+
+ t)
(1 is a density), we have
0
'I'
1
'I'
1
(1(1 + 1)
+
1)
for 0 I <s. This derives further interest by comparison with the classical moment inequality
e'f(e)de)'
I <s which also emerges by total positivity arguments in which the relevant second order determinant is of the opposite sign.. For another den: for 0
vation of (42), see 181. 4. Convexity preserving properties. The significance of the variation diminishing properties of sign regular functions described in Theorem 5 is best emphasized by drawing some of its consequences. These results are of interest in themselves and useful in applications. We consider the transformation (21) under the same conditions stated earlier. PRoPosITIoN 1.
Suppose ,.K (x, y)dp(y)
1 for all x X.
If K is SR2 and f(y) is monotone, then (43)
g(x) =
y)f(y)dp(y)
TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 343 is monotone. PRooF.
(a)
Let a be any real number and consider the relation g(x) — a
=
K(x,y)[f(y) — a]d1z(y),
x€X.
For any a, according to the hypothesis, f(y) — a changes sign at most once. The variation diminishing character of K implies that g(x) — a enjoys the same property. This is tantamount to the monotonicity of g(x). PRoPOsITIoN 2. In addition to the hypothesis of Proposition 1, we have: then y(x) (i) If K is SSR2 and f(y) is monotone and not constant (a.'. is strictly monotone in the same sense as f. (ii) Suppose X is an open interval and K is E.SR2(x). Assume that differenti. ation under the integral sign in (21) is justified. If f(x) is monotone and not constant (a.e. p), then g'(x) never vanishes. In the following proposition, the hypothesis is strengthened and now involves sign regularity of order 3. PROPOSITION 3. Let X be an open interval of the real line or an interval of integers (i.e., a consecutive set of integers). Similarly for Y. Assume that
xcX, a > 0.
y)dp(y) = ax + b,
Let K(x, y) be TP3. If f(y) is convex (concave) then g(x) is convex (concave).
PROOF. The proof is based on the identity (45)
g(x) — c1x— c2
=
K(x,y)(f(y)
—
c1y/a
— c2
+ cibla]dIL(y)
for any real c1 , cz. The function in brackets under the integral sign changes sign at most twice, since f is convex (concave). Hence, by Theorem 5, part (a), g(x) — c1x
— c2
changes sign at most twice and if twice in the same
direction as the function of the integral in (45). Thus in particular every line intersects g at most twice. Whenf is convex and f(y) — c1y/a — ca + c1b/a displays the arrangement of signs +, —, + (which is the only possibility of two sign changes) and if g(x) — c1x c2 also shows two sign changes then they occur necessarily in the sequence +, —, +. This implies that g is convex. Had we assumed in Proposition 3 that K is SR3, then we could have concluded
only that g is either convex or concave and which it is requires further discussion.
If we strengthen the hypothesis, the conclusions are correspondingly stronger. Thus PROPOSITION 4.
Assume the conditions of Proposition 3 hold.
(1) Let K(x, y) be STP8. If f(y) is convex (concave) and not linear (a.e. ia),
SAMUEL KARLIN
344
then g(x) is strictly convex (concave). (ii) Let X be an open interval;
K(x, y) is ETPS and differentiation under the integral sign in (21) is justified. if f(y) is convex (concave) and not linear (a.e. /2), then
g"(x) >
0 (g"(x) < 0).
A generalization of these assertions can be achieved by introducing the concept of convexity of order k. Explicitly a function 1(y) defined on an open interval Y is said to be convex of order k if, for every polynomial Ph_i(y) of
degree k —
1
with positive leading coefficient, f(y) —
displays at most
k sign
changes as y traverses Y.
k
changes, they occur in an arrangement ending with a + sign. In
sign
In
the circumstance that there are exactly
particular, first order convexity is synonomous with increasing and second order convexity coincides with the usual concept of convexity, etc. If I is k times
continuously differentiable, then it is elementary that f is convex of order k 0. if and only if PRoPosrrIoN 5.
Let X and Y be- open
Pj*(x),
K(x,
(46)
for
intervals.
that
X
1,.. •,k 1, and where P1(y) is an arbitrary real polynomial of degree I whose highest coefficient is positive then Pj*(x) is of the same type. Let K(x, y) be TPk; if f(y) is convex of order k, then g(x) defined in (21) is every I =
exact
convex
of
order k.
The proof
consists of an obvious adaptation of the method of Proposition 4.
Other stronger versions of this corollary hold under the condition that K is STPf, or alternately when K is ETPr(X). There also exists a version of Theorem 7 and its corollaries for the case that X and Y are sets of consecutive integers. The following examples occasionally make use of this version. We now present three applications of these propositions. 1. Let = K(x, y) denote a density function with respect to a sigmafinite measure where x serves as a real parameter. Let denote the random variable associated with p1(y). Then Pr{Y1 A) is a decreasing function (strictly) of x for fixed real A whenever K is TP1 (STP2) and /g(—Oo, Al > 0 and 00) > 0. The proof is immediate in view of Proposition 1 and the representation
A)
=
where
_(1,
y>A.
Examples fulfilling the above conditions include all the common statistical densities.
TOTAL POSITIVITY AND CONVEXITY PRESERVING TRANSFORMATIONS 345
2. We apply the method of Proposition 4 to the study of a function which arises in certain economic and queueing models. Consider the densit-y fl—I'
(n—rn)!
K,(n,rn)=
+!
+1!+ ... 0,
otherwise
(a is a fixed positive constant). Clearly, K is a density in rn = 0, 1,2. -- for rn) is TP for each n = 0, 1, 2, - •. A trivial calculation shows that
n,rn=0,1,2,•". Weprovethat
Cl'
1!
is a decreasing convex sequence of n
2!
+1.
0.
The function g(n) possesses economic interpretation; it expresses the marginal
loss in sales when n represents the capacity available in supplying demand. The fact that it is convex is of interest for applications. Let f(n) = 1 for n = 0, 0 for n > 0; then f is clearly decreasing and convex on Y = {O, 1,2, •--}. We observe that g(n) =
(49)
m)f(rn)dp(m)
where p is the counting measure on Y and g(n) is defined in (48). A straightforward computation produces the formula rn)dp(m) = n + ag(n) — a.
(50)
Therefore (51)
(c + aa)g(n) + an — aa + b =
m)[cf(m) + am + b]d,u(m),
a > 0.
We consider only the case a > 0. (The case a ,n. Assume without loss of generality that 0€ ir,V. Clearly the sequence y,, admits a subsequence such that the sequence y,,,./Ily,,,011 converges to some point qeR" with = liqil = 1. Since [O,yM1] c I?, for all i mj, it follows that [0, oo[q 7r.Y, contradicting the fact that ,r.V is bounded. Now with Dm bounded, suppose ,r5 0 whenever 9c. and card 9
PROOF.
Since the set Z =
S: S .Y" such that let I),, =
n + 1. Helly's theorem implies that . has the finite intersection property, Then whence the same is true of the family of compact sets (K n D,., : Ke * 0 and the proof is complete. of course ir.
In a study of the structure of families of convex sets with empty intersection, the families of open halfspaces are of special interest, both intrinsically and because of the following fact (Gale-KIee [1]): 1.2.
Suppose (K0: a€
is a family of convex sets in R' such that
K0
=
0. Suppose further that each set K0 is open, or each is closed and admits neither asymptote nor boundary ray. Then there is a family of open halfspaces (J0: a A} such that K0. J0 = 0 and always
We next state Klee's extension (31 of Helly's theorem to compact families of open convex sets. (A related compactness condition appears in a covering theorem of Karlin-Shapley [11.) 1.3. Suppose is a family of open convex sets in such that mt Ce = whenever C is the limit of a convergent sequence of members of If n + 1. 0, then ir5 = 0 for some .9c .5t with
A theorem of Ramsey [1] asserts that if all r-membered subsets of an infinite set A' be divided into a finite number of classes, then there is an infinite subset Y of X such that all r-membered subsets of V belong to the same class. Combining this result with Helly's theorem, R. Rado (11 has proved: 1.4. If is an infinite family of convex sets in there is an infinite subfamily .3/' of .2" which has one of the following two properties: .3/ has the finite intersection property;
for some 5 n it is true that each j members of .3/' have
inter-
section but each j + I members have empty intersection.
In any discussion of intersection properties of convex sets, the semispaces should be mentioned. When P is a point of a linear space E (of dimension a maximal convex subset of E — p. When S is such 1), a a semispace, its reflection 5' in p is also a semispace and E—. {p} = S u S'. The semispaces were introduced by Motzkin [1) and Hammer [1], who showed
INFINITE-DIMENSIONAL INTERSECTION THEOREMS
351
that in the following sense they form the unique minimal intersection base
for the class of all convex sets; 1.5. Every convex set is an intersection of semispaces. If a semispace S is the intersection of a family of convex sets, then Se .X.
A detailed discussion of semispaces is given by Hammer [1; 2] and by Klee [6), who proves the following intersection theorem: 1.6. For a convex set C in a linear space of countable dimension, the following two assertions are equivalent: C is the intersection of a countable family of semispaces;.
whenever .2' is a family of convex sets for which n.5r = C, then ,r.9' = for some countable .9' c
C
From 1.6 it follows (in a linear space of countable dimension) that if a family of convex sets has empty intersection, then so has some countable subfamily. While this property is characteristic for spaces of countable
dimension, the same conclusion holds in more general spaces when the convex
sets are subjected to suitable topological restrictions. A result in this direction was established by Klee [6] (1.8 below), and Corson has recently formulated the underlying topological conditions. For a family .9' of subsets of a topological space, let = ci n.5'. For a family let = countable .9' c Then Corson's result is as follows: 1.7. Suppose X is a topological space and W is a family of subsets of X such that X %'° has the Lindelof properly and ,r.9' is different from 7cW for each countable 3.' c Then W° is a proper superset of ,r4' and U n W° is dense in for each Uc W. PRooF.
Since {X— ..V*: countable .9' c W} is an open covering of X— fe'°,
there is a sequence , 9', , --• of by the Lindelof property of X— countable subsets of such that X— ,.V). With = = we have
U
= ci ir.5r = cI
c
c
=
=
jf = then = ci whence = Since is a countable subfamily of this contradicts the hypothesis and shows that *° is indeed a proper superset of For each whence
U€W,
(Un 'l'i°)=cl(Uri whence ci (U n 1.8.
=
and
the proof is complete.
Suppose E is a complete separable metric linear space and W is a
family of convex subsets of E such that is closed. Suppose further that each member of W is closed, finite-dimensional, or open relative to the smallest closed fiat containing it. Then ,r.9' = irW for some countable .9' c Yl. PROOF. Suppose the conclusion is false, whence 1.7 implies the existence
352
VICTOR KLEE
For each z e ?I°, let D, denote the union of all = open line segments in W° which pass through z (and D, = lx) when is a complete separable metric convex set, it is known that {x}). Since such that A is dense in ?I° (see Klee (5]). Since there exists z and it is easy to that D, is is closed we may choose ye Jx, z( is Now consider an arbitrary Ue %', and note that Un dense in When by 1.7. When U is closed, then of course dense in U is finite-dimensional, then '//° is finite-dimensional and y must be a relawhence y U. Suppose, is dense in tively interior point of W° (since finally, that U is open relative to the smallest closed flat F containing U. If y U, then y lies in a closed hyperplane H relative to F such that H misses U. Since U n is dense in by 1.7, W° cannot lie in H and hence must intersect one of the halfspaces (in F) which is bounded by H. Since is dense in must intersect the other open halfspace also, whence U intersects both open halfspaces. But then U intersects H and the contradiction implies that yE U. We have now proved in each of the three cases that y e U, whence y and the contradiction completes the proof of the theorem. Theorem 1.8 actually holds for a somewhat more general class of sets, as described by Klee (6]. While separability of the containing space plays an important role by way of 'the Lindelöf property, it is unclear whether completeness is essential. In particular, the following problem appears to be open: If is a family of open convex subsets of a separable normed linear is closed, must there exist a countable subfamily ..V of W for space and = When (or, a fortiori, has nonempty interior, an which affirmative answer follows by the above method; when = 0, the answer
of a point xe
is apparently unknown. 2. Intersection theorems involving the weak topology. Among the infinitedimensional relatives of Helly's theorem, those of the most significance for
functional analysis deal with intersection properties of closed convex sets. These results are strongly topological in flavor, but convexity enters the picture because of the fact that a closed convex subset of a locally convex topological linear space must also be weakly closed (closed under the weak topology). It is known, for example, that a normed linear space E is reflexive
if and only if its unit cell lx: t
1} is weakly compact, and hence it is x II evident that E is reflexive if and only if 0 whenever .5/ is a family of bounded weakly closed subsets of E which has the finite intersection property. Theorems of Smulian [1] and Eberlein [1] show that reflexivity and weak compactness can be characterized in terms of intersection properties of sequences of convex sets. As refiaed by Dieudonné [1] and Day [1], the result may be stated as follows: I
2.1. A weakly closed subset X of a complete locally convex space E is weakly
compact if and only tf X has the following Property: whenever K. is a decreasing sequence of closed convex sets in E and each K1 meets X, then meets X.
K,
INFINITE-DIMENSIONAL INTERSECTION THEOREMS
353
This result has interesting implications for the geometry of convex sets (Klee [1; 2], Dieudonné [1], James (1; 2], Köthe [1]) and in topological studies (Klee [4]). For the latter purpose, the foUowing intersection theorem was proved by Klee [7] and applied by him and by Corson-Klee (1]: 2.2. In every infinite-dimensional normed linear space there is a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty.
For convex sets, the following improvement of 2.1 is due to Floyd-Klee [1] and Pták [1]: 2.3. A bounded closed convex subset K of a complete locally convex sPace E is weakly compact if and only if K has the following Property: whenever H. in E and each finite intersection 11' is a sequence of closed H, meets K. meets K, then
(1] shows that if U is the unit cell of a separable nonreflexive then some continuous linear functional fails to attain its maximum on U and thus there exists a decreasing sequence f, of closed halfspaces such that hr misses U even though each set J, meets U. (The same result for the nonseparable case is stated by James 12].) Klee [8] conjectures (at Banach
least in the separable case) that this property is characteristic for weakly closed sets which are not weakly compact, and proves an intersection theorem which seems to support the conjecture. Let us say that a family of sets has the countable intersection property In line with 1.8 and 2.1, provided * 0 for each countable ..V c it is natural to wonder which Banach spaces E have the following property: 10 whenever a family of closed convex subsets of E has the countable intersection Property, then ,r.2' 0. This and related properties of E are studied by Corson [2] in a deep investigation of topological properties of spaces E., (a Banach space E in its weak topology). He considers the following properhas the Lindelof proPerty; 1110 the space E is the closed linear extension of a weakly compact set; IV° the space is normal; V° for each natural number n, the product space E is normal; V!° the space El is fraraties: 110 the space
compact. It is well-known or obvious that 110._la, II°=.VI°=-4V°, and V°—'IV°.
Corson proves that VI°=II° and conjectures in an oral communication that the conditions II°—VI° are all equivalent to each other but are not equivalent to 10. In fact, let D denote the Banach space of all bounded real valued functions
f on [0, 1] which at each point of [0, 1] are continuous from the right and have a limit from the left, with 11111 = sup (Ifx : x [0, 1]). Then Corson can show that the space Dl, lacks the Lindelof property and he suspects that D has property 1°. (By a theorem of Rudin-Klee [1] and Michael [1), E is a separable metrizable locally convex space.)
Corson's paper [2] contains other interesting results and examples concerning intersection properties of convex sets in certain Bánach spaces. For example, the space (m) lacks property and the space C.[O, 2[ actually contains
354
VICTOR KLEE
a family of closed metric cells (homothetic images of the unit cell) which has the countable intersection property but empty intersection. If C is a compact
group and L1G the space of all Haar integrable functions on C, then the indicates the topology space (L1G),,. has the Lindelöf property, where a(L1G, CC) obtained by regarding the members of L1G as continuous linear functionals on CC. Corson's most complete results concern spaces C,X, where this is the Banach space of all continuous complex-valued functions which vanish at infinity on the locally compact Hausdorif space X. 2.4. If X is met rizable, then (C0X),, has the Lindelof property hence C0X has Property 10. When X is a topological group, then of X, the Lindelof ProPerty for and normality of are all equivalent. Another interesting paper by Corson [11 contains an intersection theorem which is related to results of Berge [1] and Ghouila-Rouri [1] discussed in § 4 of DGK. For comparison, we repeat the result of Ghouila-Houri: Suppose C,, - . ., C,,, are closed convex sets in a topological linear space and each k of the sets have a common Point, where 1 k < m. If C is convex, then some k + 1 of the sets have a common Point. Corson's result is as follows: 2.5.
If W is a family of bounded convex sets covering an infinite-dimensional
reflexive Banach space E, there is a point xe E such that every neighborhood of x meets infinitely many nwmbers of of an infinite-dimensional normed Now consider instead an open covering let ö V denote the diameter of V. When linear space S, and for each V ,.oV < a simple dimension-theoretic argument shows that for each have a common point. The reasoning of finite m, some m members of Corson [1] shows that if each member of is bounded and convex, and is a subfamily of {cl V: Ve which has the finite * 0 whenever intersection property, then some finite-dimensional cube in S meets infinitely This applies in particular when S is a conjugate space many members of and F is a covering of S by open metric cells. Even when is such a special covering of Hubert space, it is unknown whether some point must
lie in infinitely many members of F. propertlea of metric cells. Completeness of a metric space 3. is characterized by the condition that if a family of nonempty closed subsets is directed by inclusion and has arbitrarily small members, then it has nonempty intersection. Without any restriction on the size of the sets, the condition is characteristic of compactness. However, there are spaces in which closed sets of a certain form always have this intersection property, irrespective of their size and compactness. The following result in that direction is due to Harrop-Weston [1]: 3.1. Suppose B is a bounded sequentially closed subset of a locally convex space E, and the closed symmetric convex hull of B is sequentially complete. Let {&}AE.f be a directed system of sets of the form BA = XA + rAB with EE
INFINITE-DIMENSIONAL INTERSECTION THEOREMS 0, where
and
A
< A' (in the ordering of A) implies BA
355
Then flAE4 BA
is a set x + rB with x = tim XA and r = urn rA. Theorem 3.1 applies to metric cells in a Banach space, where by (closed) metric cell we mean a set of the form V(z, r) = {x€M: p(z,x) r), M being a metric space with distance-function p, z M, and r 0. (Although such sets are often called "spheres", we feel that such terminology is misleading. Our spheres are sets of the form {x M: p(x, z) = r}.) An example of Sierpinski
[1] shows that the metric cells of a complete metric space may lack the intersection property expressed by 3.1. More recently,
[1] observed that
the real line can be given an invariant metric with respect to which it is complete but contains a decreasing sequence of metric cel!s with empty intersection. (Let p(x, y) = x — y I when I x — y I
when
Ix—yl>2.)
I
2 and p(x, y) =
1
+ (I x — y
I—
intersection properties of metric cells were discussed in 6—9 of the treatment there was mainly restricted to convex cells in Similar ideas have been discussed in a more general setting by several authors, Various DGK, but
some for metric spaces (Aronsza;n-Panitchpakdi [11, Gehér [1], Grünbaum [2)) and others for Banach spaces of arbitrary dimension (Nachbin [1, 2], Lindenstrauss [1]).
The expansion constant of a metric space M is the greatest lower bound EN of numbers p such that for each family { V(z,, r.) a 6 A) of pairwise intersecting cells in M, the intersection fl,6A V(z1 , is nonempty; if always V(z,, 0, the constant is said to be exact. The Jung constants
are defined similarly with respect to families of pairwise intersecting metric cells which are all of the same radius r, = r. (Of course
=
when
M is a normed linear space, but this equality fails in the general case. Compare with the definition of J(p, r) in § 6 of DGK.) Clearly EM for all r, and inequality is possible even for a two-dimensional Minkowski space (GrUnbaum [1]). On the other hand, GrUnbaum [2) describes a Banach space
M for which fit = E1 though the first constant is exact and the second is not.
A metric space M is said (by Griinbaum 12)) to have the finite intersection property (f.i.p.) provided * 0 whenever is a family of metric cells in M such that every finite subfamily of has nonempty intersection. This property is possessed by every space which is boundedly compact, and also by every Banach which is conjugate to another. Grünbaum [2] conjectures that f.i.p. is characteristic for conjugate spaces. Clearly the constants EN and are exact whenever M has f.i.p., but this is not necessary for exactness of the constants. He notes that = 2 and the constant is exact, while (c0) lacks f.i.p. even for families of cells of the same radius. The following result of GrUnbaum [2] extends a theorem of Bohnenblust [1] on projections (cf. 3.4 below): 3.2.
E1,
2 whenever M is complete.
A metric space M is called totally convex if for each pair x and z of points of M and each A €10, 1[ there exists y e M such that p(x, y) =
Ap(x, z)
356
VICTOR KLEE
and p(y, z) = (1 — A)p(x, z). The retraction constant rM of M is the greatest lower bound of numbers p which have the following property: whenever Y
is a metric space which Consists of M and one additional point, there is a transformation r of Y onto M such that rx = x for all x M, and p(ry, ry') pp(v, y') for all y, y' e Y. If there always exists such an r with p(ry, ry') rMp(y, y'), then rM is called exact.
Griinbaum [21 proves 3.3. For each metric space M, rE, while = rN when M is totally
If either constant is exact, so is the other. From this he deduces a result of Aronszajn-Panitchpakdi [1) on the extension of uniformly continuous transformations. When X is a normed linear space, GrUnbaum [2] defines the projection constant Px as the greatest lower bound of numbers p having the following property: for any normed linear space Y which contains X as a subspace of deficiency one, ,there exists a continuous linear projection P of Y onto X with p; exactness is defined in the obvious way. He proves II P11 3.4. For each normed linear space X, if either constant is exact, so is the other. convex.
He discusses the connection of this result with the extension of linear transformations and deduces Nachbin's result [1] to the effect that a normed linear space has the Hahn-Banach extension property if and only if the family of all its metric cells has the binary intersection property (cf. 7.2 of DGK). Another projection constant, is defined like above but with respect to all spaces V having X as a subspace (no restriction on deficiency). Obviously Px Px. The constant Px has been extensively studied, and in particular, GrUnbaum [31 has evaluated Px for a number of finite-dimensional spaces. Except for those involving the condition = 1 (ci. 3.6 and 3.7), the many interesting results on Px seem to have little connection with intersection properties, so will not be discussed here. The interested reader may consult the report of Nachbin [21, pp. 94—96 of Day [1], and other papers listed by them; for some more recent results and references, see Isbell-Semadeni 111. For a cardinal r 3, a metric space M is called r-hyperconvex if for each family rd,) : a A) of metric cells in M with card A < r and r. + the intersection is nonempty A V(ZG, + e) * 0 for each e > 0>. The space M is hyperconvex pro for r-hyperconvex spaces which are not (r + 1)r hyperconvex. From results of Hanner [11 (7.1 of DGK) it follows that a 3dimensional 11-space (regular octahedron as unit cell) is 4-hyperconvex but not 5-hyperconvex, and that each finite-dimensional Banach space which is • The latter statement is 5-hyperconvex must in fact be proved by Lindenstrauss [1] for an arbitrary Banach space.
In connection with 3.5, Aronszajn and Panitchpakdi ask whether, for there exist complete metric spaces which are almost r-hyPerThey produce such an example for r — 3, convex but not and show that in general, almost r-hyperconvexity implies — 1)-hyper3< r
0, the intersection V(z1, 1 + is nonemPty; whenever 3.7.
Px.. =
1
Y is a closed linear subspace of a Banach space, T is a compact linear operator
from Yb X, and that } I TI I
(1
+
> 0, I
T admits a compact extension T from Z to X such
I TI I; the preceding condition holds for dim (Z/ Y)
1.
Lindenstrauss also shows that if the unit cell of X has an extreme point, then the above conditions are equivalent to X's being isometric with a certain type of subspace of a space C(H). Nachbin [1; 2J asked whether an Banach space whose unit
VICTOR KLEE
358
cell has an extreme point must be equivalent to a space C(H). Lindenstrauss [11 supplies a counterexample, which also disproves a conjecture of Grothendieck [11. (His example is the space of all real sequences x = (x1, x2, ...) for with I Ix I = sup,, I x,, I.) On the other hand, which urn,,..,. x,, = (x1 + Lindenstrauss proves 3.8. A Banach space X is eguivalent to a space C(H) (H compact Hausdorff) the set of all extreme points of the unit if and only tf X is cell of X is nonempty, and the set of all extreme points of the unit cell of X5 is w5-closed. (None of these three conditions is implied by the other two.) Fullerton [1] defines a CL-space as a Banach space X with unit cell U such thai for each maximal convex subset F of the boundary of U, U cony (F u — F). Hanner [1J showed that every finite-dimensional 4-hyperconvex Banach space is a CL-space (but not conversely). Fullerton (11 characterizes those conjugate CL-spaces which are L-spaces or C(1f)-spaces, and Lindenstrauss [1] obtains the following characterization in terms of intersection properties:
3.9. A C
CL-space X is a C(H)-sftace if and only if are four metric cells in X such that each three have a
common point. BIBLIOGRAPHY
N. Aronszajn and P. Panitchpakdi 1. Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 8 (1956), 405-439. C. Berge 1. Sur une propriété combinatoire des ensembles convexes, C. R. Acad. Sci. Paris 248 (1959), 2698-2699.
H. F. Bohnenblust 1. Convex regions and projections in Minkowski spaces, Ann. of Math. (2) 39
(1938),
301-308.
N. Bourbaki Espaces vectoriels topoiogiques, Hermann, Paris, 1955; Chapters 111-V. H. H. Corson 1.
Collections 143-145.
of convex sets which cover a Banach space, Fund. Math. 49
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2. The weak topology of a Banach space, Trans. Amer. Math. Soc. 101 (1961), 1-15. 14. Corson and V. Klee 1. Topological classification of convex sets, these Proceedings, pp. 37-51. L. Danzer, B. GrUnbaum and V. Klee 1. Helly's theorem and its relatives, these Proceedings, pp. 101-180. M. M. Day 1. Normed linear spaces, Springer, Berlin, 1958. J. Dieudonné 1.
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W. A. Eberlein 1. Weak compactness in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 33(1947), 51-53.
INFINITE-DIMENSIONAL INTERSECTION THEOREMS
359
W. Fenchel
1. A remark on
convex sets
and polarity, Medd. Lunds Univ. Math. Sem. Tome
Supplémentaire (1952), 82-89.
E. E. Floyd and V. L. Klee 1. A characterszatton of reflexivity by the lattwo of closed subapaces, Proc. Amer. Math. Soc. 5 (1954), 655-661.
R. E. Fullerton 1. Geometrical characterizations of certain function spaces, Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, pp. 227-236, Academic Press, Jerusalem, 1961. D. Gale and V. Klee 1. Continuous convex sets, Math. Scand. 7 (1959), 379-391. L. Gehér 1. Ober Fortsetzungs- ned Approximations problem. fur stetige Abbildun gem von ,netrischen Raumen, Acta Sci. Math. Szeged 20 (1959), 48-66. A. Ghouila-Houri 1. Sur l'étud.e combinatoire des families de convexes, C. R. Acad. Sci. Paris 252 (1961), 494-496.
D. S. Goodner 1. Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89-108. A. Grothendieck 1. fine caractérisation vectorislie-mitrique des espacea V. Canad. j. Math. 7 (1955), 552-561.
B. Grünbaum 1. On some covering and intersection properties in Minkowski spaces, Pacific J. Math. 9 (1969), 487-494. 2. Some applications of expansion constants, Pacific J. Math. 10 (1960), 193-201. 3. Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465. P. C. Hammer 1. Maximal convex sets, Duke Math. J. 22 (1955), 103-106. 2. Semispaces and the topology of convexity, these Proceedings, pp. 305-316.
0. Hanner 1. Intersections of translates of convex bodies, Math. Scand. 4 (1956), 65-87. R. Harrop and J. D. Weston 1.
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M. Henriksen 1.
Seine remarks on a paper of Aronszajn and Panitchpakdi, Pacific J. Math. 7 (1957), 1619-1621.
J. Isbell and Z. Semadeni 1. Projection constants and spaces of continuous functions, Trans. Amer. Math. Soc. (to appear). R. C. James 1. Reflexivity and the supremum of linear functionals, Math. Ann. 66 (1957), 159-169. 2. Characterizations of reflexive Banack spaces, Mimeographed abstract (2 pages), Conference on functional analysis, Warsaw, 1960.
M.Jüza 1. Remark on complete metric spaces, Mat.-Fyz. Casopis. Slovensk. Akad. Vied. 6 (1956), 143-148. (Czechoslovakian. Russian summary)
S. Karlin and L. S. Shapley 1. Some applications of a theorem on convex functions, Ann. of Math. (2) 52 (1950), 148-153.
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V. Klee 1. Some characterizations of reflexivity, Rev. Ci. Lima. 52 (1950), 15-23. 875-883. 2. Convex sets in linear spaces. II, Duke Math. J. 18 3. The critical set of a convex body, Amer. J. Math. 75 (1953), 178-188. space, Trans. Amer. Math. homeomorphisms in 4. Convex bodies and Soc. 74 (1953), 5. Separation properties of convex cones, Proc. Amer. Math. Soc. 6 (1955), 313-318. 6. The structure of semi8 paces, Math. Scand. 4 (1956), 54-64.
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8. A conjecture on weak compactness, Trans. Amer. Math. Soc. 104 (1962), 398-402. G. Kbthe Topologische lineare Räume. I, Springer, Berlin, 1960. J. Lindenstrauss On the extension property for compact operators, Bull. Amer. Math. Soc. 68 (1962), 484-487.
E. Michael 1. On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961), 921 T. S. Motzkin 1. Linear inequalities, Mimeographed lecture notes, Univ. of California, Los Angeles, 1951.
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2. Some problems in extending and lifting continuous Linear transformations, Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, pp. 340-350, Academic Press, Jerusalem, 1961. R. L. Plunkett 1. Concerning two types of convexity for metrics, Arch. Math. 10 (1959), 42-45. V. Pták 1. remarks on weak compactness, Czechoslovak Math. J. 5 (80) (1955), 532-545. R. Rado 1. A theorem on sequences of convex sets, Quart. J. Math. Oxford Ser. (2) 3 (1952), 183-186.
F. P. Ramsey 1. On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286. M. E. Rudin and V. L. Klee 1. A note on certain function spaces, Arch. Math. 7 (1956), 469-470. L. Sandgren 1. On convex cones, Math. Scand. 2 (1954), 19-28. W. Sierpinski 1. Sur une definition des espaces complete, Ganita 1 (1950), 13-16. V. Smulian 1.
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(47) (1939), 317-328. (Russian. English summary) F. A. Valentine 1. The dual cone and Helly type theorems, these Proceedings, pp. 473-493. UNIVERSITY OF WASHINGTON
ENDO VECTORS' BY
T. S. MOTZKIN 1. IntroductIon. In a vector space V over the field R0 of real numbers, a convex set may be defined as a subset S of V containing 21s1 + A2s2 for all S, s2 S, and all
=1 Strengthening the requirement by deleting the restriction (1), or (2), or both, on the two-dimensional coefficient vector A = (2k, 22) E R0, we obtain, instead of the convex sets, the family of flats (linear subspaces) in case (1) is deleted, of 0-flats (flats through the origin 0 of V, or empty) if (1) and (2) are deleted, and of convex O-halfcones if (2) is deleted. Thus each of these four families 2
is defined, with respect to the "center" 0, by certain characteristic "endovectors" A, where A is an ordered pair of coefficients of a linear combination under which every member of the family is closed. The following study of sets of endovectors defining families of point sets falls into a general theory (Chapter 1) and into investigations of the translation invariant case (II), of sets of one-dimensional endovectors (ffi—V) and of other special endovector sets (VI). The results are partly of an arithmetic (in a wide sense of the word), partly of a geometric nature. Among those of the first type we note Theorems 20— 22 on the structure of translative endovector sets with two-dimensional integral generators, 76—79 on certain nontranslative endovector sets with a single two-dimensional integral generator and on proper endothety, 36-37 on centers of symmetry and autothety, 29-32 on semigroups of one-dimensional endovectors, and 27 on proper dimensions of endovector sets; of the second type, 50-51 on the structure of ambiconvex sets and 49 and 54-73 on overstar and inverse overstar centers. The present study has connections with several topics of current interest
(theory of structures, affine analysis of convex sets) but presupposes only standard material, as shown by the absence of references. Numerous points will be noticeable where the investigations have been, more or less arbitrarily, broken off and which are awaiting further research. CHAPTER I.
2.
GENERAL THEORY
FamIlies. A set S is said to include every
S
(element of S) but to
contain every c S (subset of S). This paper includes results presented to the American Mathematical Society on October28, 1961. 361
T. S. MOTZKIN
A family F, i.e., a set of sets, is called a classification of its union U F into classes if it consists of disjoint nonempty sets. Every classification of U F corresponds to an equivalence relation between the elements of U F. A family F is called intersectional if F includes the intersection of every subfamily of F. In particular, F includes the intersection of the empty family;
this intersection is the "largest" F (i.e., it contains every F) and hence is U F. Every Sc U F determines a "smallest" e F containing it, viz., the intersection of all F containing S; this smallest F is said to be spanned by S and called the hull H(S). The hull of a union of sets is their join. A family F is called unional if F includes the union of every cF. In particular, F includes the union of the empty family; this union (whether or not stipulated as empty) is the smallest e F and hence is the intersection fl F. Every S fl F determines a largest e F contained in it, viz., the union of all F contained in S; this largest F is the core of S. A family F of subsets of a set V is called complemental if F includes the complement V — S of each member SE F. A complemental family is unional if and only if it is intersectional. We shall use THEOREM 1. Every intersectional, unional and complemental family with nonempty union consists of the unions of the classes in a classification of its union.
These classes are the hulls of the singletons contained in the union. 3. Notation. We denote in the sequel by R a, not necessarily commutative, ring with unit 1, by R(1) the subring spanned by 1, and by x (the "characteristic" of R) the cardinal (number of elements) of R(1). Then Z = 2, 3, 4,-.., or
By V we denote a vector space (left module) over R: the operation v1 + v2 for points (elements) of V is an abelian group, and the operation p V, p E R, V E V. is distributive in p and in v and associative in p, with 1.v = v. The point 0 is called origin; we assume that V includes at least one point besides 0. A linear transformation T of V into itself or another vector space over R is a mapping v —* T(v) such that T(p1v1 + p2v2) =
T(v1) + Pt T(v2). A homo-
thety is a mapping v pv; it need not be a linear transformation. A translation is a mapping v —' v + v0. If S is a subset of V then the image of S under these mappings is denoted, respectively, by T(S), pS, S + v0. If the image is S itself the mapping is, respectively, a linear automorphism, an autothety, an autotranslation (period).
The particular vector space called k-space over R and denoted by R" (k =
Consists of all k-tuples A = (A,,, . . ., with coordinates R, with the usual definitions of addition and scalar premultiplication. By Uk we denote the set of the k unit vectors (the rows of the k by k 1,
2,
•..,
unit matrix) by ik their sum (1, ..., We also consider unions of disjoint vector spaces over R. The union R1 U R2 is denoted by R, and the set U,, U U2 Li-- c R by U. 4. Endovectors. DEFINITION 1. For a given subset S of V, the vector A =
ENDOVECTORS •, •,
(As,
s1,
363
is called a k-O-ENDovEcToa or S if S includes
ASj for all
S is said to be ENno-A or to have A-CENTER 0.
Similarly if S is a subset of a union U V1 of disjoint vector spaces we call A an endovector if S includes A1s1 for all s1, •, e S for which this sum has a meaning. For any ye V, O-endovectors of S are called v-endovectors of the translated set S + v. The vector A = (A1, •-, Ak) is a v-endovector of S if and only if v + A'(s, — v) E S for all s1, •, 5. If v becomes infinity (topologically or projectively) the condition becomes w 1s S, A = 1; for k = we call w an endotranslation (hemiperiod). 1
A
THEORFM 2.
The set of 2-centers (A =
ifSisendo.A*, A*=(Al,...,Ak,
(As,
..., A'))
of S contains S if and only
For either condition amounts to saying that S includes s +
—
s)
for
seS.
all
For any set 4cR,2 a set Sc V (or c U V1) such that every AcA is an 0endovector of S is said to be endo-A or to have A-center 0. The set of A-centers of S contains S if and only if S is endo.4*, where A* (defined as in Theorem 2) 4* if A e A. The empty set 0 is endo-A. The set of A-centers of 0 is V. Immediately verifiable are: 3. if S is endo-4 so are pS, if pA, = A,p for every coordinate 2 of every A €4, and T(S) for every linear transformation T. (For S + v see § 12.) THEOREM 4. If S1 and S2 are endo•A so is their sum + S2, and, in particular, their direct Product S , 0) + (0, S2). = The sum S1 +S2 of the sets S1 and S2isthesetof ails1 +s21s1eS1,s2€S2. 5. A-hulls. Obviously the family a(4) of all endo-A sets in V (or in U V1) is intersectional. The hull of S c V (or c U V1) in this family is called the
We have SEa(4) if and only if H4(5) = S; if and only if A•S c S, where 4-S is the set of all A,s5, s, ES, A €4; and if and only if S' = S, where S' = S'(A) = S U 4-S. A-hull H4(S).
THEOREM 5.
(1) (2) (3)
H4(S)=SUS'tJS"U.-.; S'(HA(U1)) U S'(H4(U2)) U = S'(J-fA(U)); HA(S) = Us1 H4(S1), where S1 runs over all finite subsets of S. HA(S)
Indeed, H4(S) consists of all 2L1L2- --•
•L1 s with A
4 U U, L1, -
, L matri-
ces whose rows are A U U with interspersed zeros, s a column with elements in S. Subdivision according to j or to the number of elements of s yields (1) or (2), respectively. From (2) follows (3) in view of H4(S1) = SI'(H4(Uk)) for any S1 with at most k points. Instead, the theory can be developed for ,1 c R'. the set of sequences over R with tinitely many nonzeros.
T. S. MOTZKIN
can , If repetition of elements in s is permitted then the matrices L,, if A In addition, be required to have only one nonzero in each column.
includes a vector with sum unity and not a unit vector, then the rows of L,, •, L can be required not to be unit vectors. On the other hand, to obtain each e H4(S) but not with minimal j it is sufficient to use matrices all of whose rows except one are unit vectors. 6. Complete endovector gets. DEFINITION 2. A set A c R is called COMPLETE if there exists a set S such that A = A(S), where A(S) denotes the set of all endovectors of S. THEOREM 6.
The statements:
(1) A is complete; (2) A=J-b(U);
(3) A=A(A), are equivalent.
Indeed, (1) (2) expresses that a set is complete only if it includes all L, defined as before, classified according to the number of columns
of L. Further, if (2) holds, then every AL is an €A (i.e., Ac A(A)) and,
since A contains U, an endovector of A must lie in A, whence (3). Finally is trivial. We note specifically these properties of complete sets: (1) Every complete endovector set contains the set of all unit vectors. (2) Every complete endovector set is permutalionally symmetric (for every matrix of A permutation ir,, •, of 1, . . k we have A = (A,, . .., e A. permuted unit vectors = (Art, ..., O)e A. A is equivalent to (A,, .. , A —o (A, + A,, (4) The elements can be compressed, i.e., (A,, . .., Ak) e A. (To every classification of (1, . . ., k) belongs a compression of A.)
(3) Zeros are irrelevant, i.e., (A,, .. . ,
.
If A is complete and includes 0 (in some Rk), then (A,, . .., A,,÷,) e A
(A,, ...,Ak)EA ("suppression").
If A is complete then H4{1} is the set of sums Z
A e A, while HA{0, 1)
is the set of coordinates A, , A E A. If Ak(S) means A(S)flRk then the sets Ak(S) can be characterized by = Ak. The set Ak determines 4k-1, but not conversely. The k by k matrices L with rows in Ilk form a multiplicative semigroup, there correspond different semigroups. In particular, to different sets A, is a multiplicative subsemigroup of R. Also the w by w matrices whose rows are infinite sequences consisting of an e A and zeros form a multiplicative semigroup M(A). Denoting by M_ the set of all w by w matrices whose rows consist of one 1 and zeros and by M.,. the set of all w by w matrices with rows R" we have: THEOREM 7.
A set M of w by w matrices is an M(A) for some complete
endovector set A if and only if M is a multiplicative semigroup and M. c Mc M+. 7.
Intersection of complete endovector sets.
THEOREM 8.
The intersection
ENDOVECTORS
of complete endovector sets is complete.
contains U and AL, where For if the sets 4( are complete then A = and therefore in A. A and every row of L belong to A, is in every The complete endovector sets form thus an intersectional family, and every set A c ,k has a complete hull H(A). THEOREM 9.
We have H(A) =
114(U); hence 114(S) =
S'(H(A)).
For HA('U) consists of all ALLL L, and is therefore complete; hence H(A). On the other hand, by (2) in Theorem 4, H(A) = Hfl(4}UD !14(U).
H4(U)
If A = H(Ak) for some k, the smallest of these k is called the basic diménsion dim A of A; if no such k exists we write dim A = For an example with k = 3 see § 14. 8.
Complete families.
DEFINITION 3.
A family a of sets S c R is called
COMPLETE if there exists a set A c R such that a = THEOREM 10. There is a 1-1 correspondence between complete endovector sets and complete families, given by a = a(A) or equivalently by A = fla A(S).
For firstly, if a = c(A) for some A c R then A c A(S) for every Sea, hence also H(4) c A(S), hence a = a(H(A)). Secondly, A c fl,A(S) and if A is complete then A(A) = A, hence Ac a(A), hence one A(S) (viz., A(A)) A, hence A = flaA(S).
The correspondence has the duality properties of being antimonotone if a increases, A decreases, and of associating intersections with joins. It goes without saying that also the class of complete families is intersectional, and thus every family a has a complete hull H(a). Clearly 11(a) = u(A), where A = flgA(S). 9. The four main families. To U and the smallest and largest complete endovector set, correspond c( U) and a(R), the largest and smallest complete family; a(U) is the family of all sets s c R, while a(R) is the family of 0-fiats. Note that U = H(Ø), a(U) = a(Ø), while R = H(Rt), c(R) = c(R2). In this statement, R2 can be replaced by the set of all (A1, 1) or by R' U {(l, 1)) (or, if R is a division ring, by R' U {A}, with any A with more than one nonzero element). The 0-flats in R1 are the (left) ideals. The third important complete endovector set, A(UI),' consists of all vectors with sum unity; a(A(Uj)) is the family of flats. The straight line through two points v1, (v1 * v2) means their flat hull v2}. If R is a division ring other than {0, 1) then A(U1) = H(A2(U1)), a(A(U1)) = a(A2(Uj)); for R = {0, 1), replace A2(U1) by {(1, 1, 1)). Finally, U U (0) is complete (where (0) is the set {0 in R', 0 in R', . . and a(U U (0)) = u(0) is the family of 0-sets (sets empty or containing 0). As (1,1) is in R but not in U, UiJ (0) or A(U1), these three differ from R and hence (cf. proof of Theorem 12) from each other; the latter follows also by noting that 0 is in UU (0) but not in A(U1), and that (1, 1, —1) is in A(U1) but not in U. .
T. S. MOTZKIN
366
THEOREM 11.
For R =
(0,
1) the sets, fiats, 0-sets and 0-flats are the only
complete families.
For by permutational symmetry and irrelevance of zeros a complete ii in R, for R (0, 1), is given by stating which vectors
endovector set
(1, •••, 1) Rk, = 0 e Rt, belong to it. But ltk+1 4, k 1, implies lzk+I + (1,0, . . ., 0) + (2k — 1)(0, 1, ., 0)€ 4, hence 12*_I €4, and similarly Likewise '2k A, k 1, implies l2k + (2k — 1)(1, 0, - -,0)e A, hence l2k+s A. e A, and similarly A; also 121+1 + (2k — 1)(1, 0, . •, 0) A, hence h, A. '2k—i or , k 0, or of L and Thus A consists either of all or of all Another proof goes via Theorem 79. only of Q.e.d. lk
•
10. Complete classes. DEFINITIoN 4. A class C of complete endcwector sets, and the corresponding class of complete families, are called COMPLETE if C includes the intersection and the join of every subclass. The totality of complete classes is intersectional, and every class of complete
endovector sets or families has a complete hull. THEOREM 12.
The families of sets, flats, 0-sets and 0-flats form a complete class.
For obviously it(U1) fl (U U (0)) = U; on the other hand H(4(U1) U (U U (0)) = 11(4(U1) U (0)) = since every 2 can be lengthened to be an E 4(U1). Hence the join and intersection of the families of flats and 0-sets are the and 0-flats. 11. Gapless classes; maximal and minimal families, DEFINITIoN 5. A class C of complete endoveclor sets, and the corresponding class of complete families, are called GAPLESS if 4' C, A" e C, A' c 4 c A", A complete imply A C. The totality of gapless classes is intersectional, and every class of complete
endovector sets or families has a gapless hull. DEFINITIoN 6. A complete endovector set 4 * U is called MINIMAL if (A, U) is gapless; a complete endovector set A * R is called MAXIMAL if (.4, R) is gapless. The corresponding complete families a(A) are called MAXIMAL and MINIMAL,
respectively. THEOREM 13. The family of 0-sets is If and only if R is a division ring the family of flats is minimal. Indeed, UU (0) is minimal. Further, if the complete endovector set A contains .4(U1) and includes some A not in 4(U1) then, if R is a division ring (other than (0, 1), in which case see Theorem 11), A contains the straight lines through A and points of .4(U1), hence (repeating this argument once) all points of R. On the other hand, if R is not a division ring then R contains an ideal R' different from R, (0) and 0, and the set A of all 2 with E 1 while 4(U1) cA cR, A * 4(U1), A R. (modR') is complete (cf.
CHAPTER II.
12.
Translatlve families.
TRANSLATIVITY
A vector v
in
V such that S€u(A) implies
ENDO VECTORS
S + V 6 a(A) is a period of u(4). DEFINFrION 7. A family u(A) in V is called V'-TEANSLATIVE if V' gs the set of periods of o(A).
This is equivalent to saying that, for 2 A, S c V, if 2 is a v-endovector of S for some v, then A is a (v + v')-endovector of S for all v' V'. THEOREM 14,
The statements:
(1) o(4) is V'-translative; (2) (3) {v'}€c(A) for every
VI,
are equivalent.
eS for 2€ 4, s, ES implies v' + A(s, — v') S For if (2) holds then S, v' V', i.e., (1). Every family e(A) includes the set {O), hence (1) implies (3). If (3) then A V-translative family g(4) is called translative; this is equivalent to saying for A e A, s2
that, for every S c V. the set of A-centers is either 0 or V, or that Se e(A) implies that also every affine transform T(S) + v o(4), where v V and T is a linear transformation. 13. Translatlve endovector sets. In particular, a(A) is translative if Ac 4(U1). A family a(A) in I? is translative if and only if A c 4(U1), i.e., if c(A) includes the flats. Subsets of A( U1) will therefore be called translative endovector sets. The complete translative endovector sets form the gapless hull of { U, 4(U1)). For complete translative A, the sets U, U U (0), 4 and A' = H(A U (0)) form a complete class. The family a(A') consists of the 0-sets in o(4); conversely, given a complete endovector set A' (0), the family a(4), where A = A' fl A( LI1),
is the complete hull of the translates of the sets in a(4). The translates themselves form a complete family if and only if A' = H((4'
fl
U (0));
this is not always the case. In fact, the H(4 U (0)), 4 complete translative, are only for R = {O, 1) the gapless hull of {U. U (0), R}, as shown, e.g., by A' = H(R1), the set of all A with at most one nonzero (cf. § 21): here A' fl 4(U1) = U, H(UU (0)) = UU (0) * A'(R * {0, 1)). 14. Gilds. For every subring R' c R with 1€ R', the set of = Uk vectors A with all e R' and the set 4(U1) fl are complete; the corresponding
complete familes are the 0-R'-flats and the R'-flats. Exactly as in the proof of Theorem 11 we see that The 0-R'-flats are fl 4(U1)) U (0)) = thus the 0-sets in the translative family of R'-flats. For the smallest subring, viz. R(1), we obtain the 0-grids (additive groups or 0) and grids (0 or cosets of additive groups). THEOREM 15. The complete endovector sets corresponding to the family of 0grids and to that of grids are H{(1, 1)) and H{(1, 1, —1)), respectively. = H{(1, 1)). Further if For obviously R(1) c H{(1, 1)) whence easily
let
A'=
T. S. MOTZKIN
368
= (1, 0, •, 0); then A' e A, u1 A, A ÷ is1 — A' A, hence 0, 1 — Ak, Ak), A =(2 + is, — A') + A' — is, H{(1,1, —1)) if (1 —Ak,Ak) and A + is1 — A' are e H{(1, 1, —1)). But A + is1 — A' belongs to a smaller k, and we end up with (0,
•
Now (2,—1)e H{(1,1,—1)}; hence also (3, —2)=2(2,----1)—1(1,0)€
1, —1)),
H{(1, the
etc.
to
Another proof translates an endo-(1, 1, —1) set
0;
then
set becomes endo-(1, 1).
Note that H{(2, —1)) = H{(1, 1, —1)} only for odd X (see § 17). The autotranslations of an arbitrary set S c V form an 0-grid ST. S is a grid then is a translate of S.
If S is
an 0-grid then
The
periods
of an arbitrary complete family o(4) form an 0-grid.
15. Convex sets. For every subserniring (subset closed under addition and multiplication) R' c R with 0 c R', 1 R', the set A of vectors 2 with all A eR' A, = 1 is complete, and A = H(A fl Rt); c(A) is the (translative) family and of R'.convex sets, and HA(S) is the R'-convex hull of S. The family of R'convex 0-sets is o(A'), where A' = H(A U (0)) consists of all A with A1 R', 2, R' (these are the A that can be lengthened to be an e A; the com1 —
pleteness If R'
is easily verified).
is a ring the R'-convex sets and 0-sets are the R'-fiats and
For the smallest subsemiring, R'(1), we obtain for finite R(1) again (since = R(1) in this case) the 0-grids and grids, for infinite R(1) (when A = U)
R'(l)
the 0-sets and sets. 16. If S is R'-convex so is Indeed, if s€S s + v,eS and plies s + X21v, = +
-
if
A,eR' and
=
then seS
1
im
co,nplete hull H({,a}), p = 1, has the coordinates 1), and of all vectors obtained from the by Permutation, compression and insertion of zeros. Allowing suppression we obtain H({p} U {0)). 16. SIngle generator. THEoREM 17.
consists
of all p"', 1
0, 1,2,
-
The
-, where
This follows from Theorem 9; (1), § 5; and the next to last remark in § 5. Denoting by H1 the set consisting of p" and of the vectors derived from
we have suppression
THEOREM 18.
elements,
H,+,,
H({p))
H0 U H1 U - -. -
Similarly for
H,'.
where
is allowed.
+p = 1,
The set R2
but no more; for R1 fl
Indeed, p" has
H
coordinates
may have
the maximal number is
() coordinates
-
-. (note
that
ENDO VECTORS
= pp,) and each A R2 n H. is given by its first coordinate A,, which may be any partial sum of the coordinates of p", viz. A, = + •.+ where
e0=0,1;e, =0,
Similarly AER°flHj' is given by
(1) o,e, + €, , A, = (2) = , Thus R fl H, has 2 elements (1,0), (0, 1) for 1 = 0, at most 2 others , p,) for 1 = 1, at most 8, 52,636, 16724, 1036272, --• others for I = (P, , ps), 2, 3,4,5,6, •••. The corresponding numbers for R2 fl H.' are 3, 6,45,846, 55800, for 1 = 0, 1, 2, 3, 4, 5,The maximal numbers for Rb fl H, and 1? fl H!, p = (p,. - - -, Pb) with comk)+ k) where k)+k — 1) and muting Pb, are similarly seen to be 17232264,
the products are extended over all k-nomial coefficients (I; k) = 1
=
For k
to n" and (,i + 17.
3
and noncommuting Pt the exact bounds are increased
1)14.
SIngle Integral generator. THEOREM 19. For p = (—1,2), H({p)) consists For E R(2) for every j * R(1) for one = 1,
of all A with E
H({.p}U(0)), omit ZAi=1. Firstly the condition is necessary since, mod 2, H = H({p)) is U. Secondly, (2, ,1
—A)cH,(A,—1,2—A,)eH—.2(A,,1—A,)—(A, —1,2—A,) =(A, +1,—A,)eH;
R(2) ( = 2, -• •, n; hence (2,, 1 — A,) e H for all A, R(1). Finally if 2, R(1), n 3), and if every such 2 for n — 1 belongs to H, then (A,, •-•, 2,,) = 2(0,- -,0,1 —A,,/2, 2,,/2)—(—A,,-- -, —A,,_2,2—2,,—A,,_,,0)EH. The second con-
tention follows by suppression and completeness check. If is (finite and) odd then R(2) = R(1). H,, THEOREM 20. For p = (ji,, 1 — p,), R(1) infinite, p,e R'(l), p, * 0, 1, (A, , Furthermore, or HI, with maximallA,l +12,1, we have IA,! +1211 =12p, — 1!'. = 1,2, 0, 1,p, or 1 —ps (A,, 22)6 H({p)) if and only if A, ER(1), 23€R(1), A, + + 2, = 1 and writting A,, A,, (mod p,(1 — p,)); for H({p} U (0)), omitting A, + A, instead of A, gives a necessary condition.
The first statement follows from (1), § 16 by selecting = 1, = 0, e, = = 0, --•; (2) yields nothing larger. Secondly, the set of (at most) four (h), residues 0, 1, p,, 1 — p, (mod — p,) is easily seen, by 16 trials, to be endo-p. Further if 4(0, 1) includes p,, 1 — p, , p,(1 — p,) and p,(1 — p,) + 1 then (as seen by substituting the latter two or 1,0 for 0, 1 and repeating) it includes Ap,(1 — p,) + A', 2' = 0, 1,p,, 1— p,, for every AER(1). Clearly p,, 1— p, and 4(0, 1). We shall now show that p, — + 1 is of p,-0 + (1 — p,)-p, are the form (1), § 16, with 1 = p,. Write 1 as E — p,)t and — as Hence — p,)1; the coefficients are then too large for (1). = subtract from terms k and k + 1 equal and opposite amounts, viz.
T. S. MOTZKIN
370 —
1),4"'(l —
and YkPi
Pt.
k =
1,3,
and
the set
5,
=0 of
•,
—
where'
= 2,
This
•
, p1
works for all
four; for Pt
1)])
—
]
2 for odd p1 and k = 1, 3, 5,
for all other k.
nine (for
—
([(lit — 2)/a
=1+ for
(1
— 3,
p1 — 2 for even Finally
1) e R'(l).
= 3, twelve)
residue pairs complying
the last condition of the theorem is easily seen (by checking the residues A,, + A, separately) to be endo-p and endo-0; but Theorem 19 shows of that the condition is not sufficient. 18. Other Integral cases. THEoREM 21. For R = R(1), a set A c A,(1) is a 4, if and only if the set A' of first coordinates of elements of A is the union to
multiplicative semigroup F of idempotent residue classes e R(l)IR(') of a containing 1 — r with r. onto R(1), finite, preserves hySince a homomorphism of R(1), x = and conclusion, we may assume x = Because of the translativity of A and by Theorem 20 it follows that the set of A such that A A', A + I €4' R'(I), and A' is the union of a set F of consists of all multiples of some idempotent residue classes R(1)/R(v). For A = A,(4,) it is necessary and sufficient that 0€ F, 1€? and that F. F implies + (1 — i',, pothesis
Setting Ti=O,Ti=l we have 1—r,EF; setting we have rireef'. On the other hand if F is a nonempty multiplicative semigroup of idempotent residue classes containing I — r with r then i(1 — r) = 0€ F, 1 — (1 — (1 —
—
— r.(l
+ Ts
Ti))
= T.Ti +(1
0
= 1 e F, and
T*Ts)(l + TiTs — —
22. For x = 1' for the number of such sets A and given v is 1 for ii = 0 (4 = U2) and i' = 1 (4' = R(1)); for v (decomposition into primes), it is the number c, of classifications of a set of k = k(M) elements. Every A belongs to only one M 0, except that if A belongs to an odd v it belongs also to For finite X, the total number of sets A is
where c0 = 1.
Indeed, r(l — r) = 0 shows that if those prime factors
of r for which j €1., c (1,
of
divide
every
the other pj divide every element of 1 — r and determines r uniquely. Since = f1.1 U I.,,, F corresponds to a complemental and unional family of subsets of {1, .., k) and therefore by Theorem 1 to a classification of (1, •, k), whire every classification of {1, •, k} leads to a family of subsets and to a semigroup F with the required properties. The cardinal of F is even and divides 2*. If 4 belongs to more than one R'(l) then t'0, the smallest v of A, is the smallest A(*0) R'(l) such that 2 £ A, A + 1 A; and any other v of A is a multiple pu,, p 2. Since — a',,) 0 (mod pv0), we have 1 — t'o 0 (mod p). But also 0 (modpv,), ml 0 (modp); hence p element
*
means largest integer.
•, k)
371
ENDOVECTORS
(mod 2). Every A belonging to an odd v. belongs also to 2v0, since 2(1 — Q.e.d. (mod implies 2(1 — A) 0 (mod
2)
0
We have Ck = E C*.J, where 4.j is the number of subdivisions of {1, - -•, k} = into j classes; Ck,J are the Stirling numbers given by c*., = 4.k = 1, •-• begins 1,2,5, 15,52,203, 877,. sequence c1 The Ck-lJ_1 + JCI-L.J.
begins 1,2,2,2,3,2,3,3,3,2,6,2,3,5,4,2, •••. The sequence 19. Real eases. For R = R,, the field of real numbers, we restrict ourselves of nonnegative numbers, to closed A,. Letting R' in § 15 be the semiring is a flat. we obtain the convex sets and convex 0-sets. For convex S, TNEOREM23. ForR=R, completeAc 4(1) and closed A,,A 1, and similarly if 1 — I> 1, which must hold for some k = 1,2, if without For AcUsee Theorem 23 and its proof. (2', 1 — A) and ((1 — A.)*, 1 — I
CiurrEa III. UNIONAL FAJIIUES 21. Cones. The sets H(R') and H(R' — {O)) consist of all A with at most one or, respectively, exactly one nonzero; dr(R1) and o(R' — {O)) are the families of strong 0-cones and of 0-cones. They differ from each other and from the four families in § 9 if R is not {0, 1}; in the latter case the strong 0-cones and 0-cones coincide with the 0-sets and sets. According to the parenthetic statement in § 9 the family of strong 0-cones is minimal if R is a division ring. 25. For R = (0, 1, —1) the sets, 0-sets, 0-cones, strong 0-cones, flats
T. S. MOTZKIN
372
and 0-fiats are the only complete families.
Here a complete endovector set A is given by enumerating the vectors lk,1 (k ones, 1 minus ones; 10.. — 0) in A. By adding k — 1 and subtracting l
1, for k—lEO (mod3):
units at a 1, a —1 or outside, we get from and Ikd÷l; for k —
1*—i.,, 1
l
—1 we get lh—ia+1,
and subtracting k—i units to
we subtract (k
1), 'k—i.l+* (k
1) e A implies
k
we get for k—la —1:
for k — 1 0: ik—1,l+t, ik.l—a, For k — 1 1, 1 1, from k positive and L — 1 negative units and repeat, obtaining
all Ii,, with h — I
2), ik+1,1—l,
ik—i.l-I ,
Since ik (k 4,
C A, it follows that + 2 (at a 1) + (k — 2)(outside) = 1, except imply each other and Since 1.,, (1 3,
0) implies 1 and —11.0 —2 (at a — 1)— it follows that every (1—3) (outside) = with k — I 0 or —1, except 1, and 10,1, implies all 'k, The contention follows now easily. Another proof 1
goes via Theorem 79. 22.
Endovector sets of basic dimensIon 1. THEoRBII 26. For a complete
endovector set A, these four properties are equivalent:
(I) dimA=i;
(2) a(A) in R includes the strong 0-cones; (3) a(A) in R is unional; (4) the proper dimension of A A is at most 1. The Proper di,nension of a vector (As, •--, is the number of its nonzero coordinates. The equivalence of (1), (2) and (4) is immediate. If (4) holds then HA(S) = Uses HA({sJ) (cf. (3), § 4), hence (3). If (3) holds then R' e c(4) implies Ha(Uk)Eo(A), hence Ak C HR(Uk), i.e., (4).
The proof shows that, for a set A of basic dimension 1, o(A) in any V is unional; and, of course, o(A) in any V includes the strong 0-cones. The sets of basic dimension 1 form the gapless hull of { U, H(R')J. Among them, only U is translative. Proper dimensions of endoveetors. Condition (4'): the proper dimension of A e A is bounded, is not always equivalent to (4), as shown by A = H(2, 2), 23.
It is if R is a ring without zero divisors: Tuzoanu 21. If R is a ring without zero divisors then the set of proper
X = 4.
di,nensions of vectors A
A, where A is complete, is one of
the last onlyforx=2,A={11,i.,---). For if (A,, A, 2,2. * 0, then (At, 2122, 22) C A; similarly •we see that A includes vectors of arbitrary positive proper dimension. On the other hand (2k, '',Ak)€A, ff2, * 0, k 3, implies (A,,..., + 21)EA which is of proper dimension k — 1 except if Ak_i + Ak = 0. If the latter holds for any two coordinates of A then X = 2, 2 = and 2,1.-, A. Hence either , e A
ENDOVECTORS
e 4 which implies (At, (see before) or (As, that A includes a vector of length 2 except if for every implies 24. Endothetle aeta.
it follows + , + = 1 and not 0 €4. Finally
The sets 4 of basic dImension 1 correspond biuniquely
to the multiplicative subsemigroups G' c R with 1 eG', via G'= A fl R, 4= H(G'); u(G') = o(4) is the family of 0-G'-endothetw sets (en4othetic = into (self )homothetic).
For the smallest subsemigroups G' except (1), viz. G'(A) (the subsemigroup 1.2, A', -.. spanned by A), we obtain the 0-2-endothetic sets. (To be exact,
the minimal G' 0-sets.
1 may be only some of the G'(A)) For A = 0 these are the
If A has an inverse A", then S c V is 0-A-endothetic if and only if the compl(ment V — S is 0-A1-endothe*. sets. THEOREM 25. The comPlete endovector set 4 gives rise
to a untonal and complemental family a(A) in R if and only if A = H(G) for multiplicative group G c R with leG. For only if the subsemigroup G' c R, 1 e G', is a multiplicative group does A'€H0.(A) imply
If G c R is a multiplicative group with 1 e G then e(G) in any V, the family of 0-G-autothetic sets, V is unional and complemental and consists therefore
by Theorem 1 of the unions of classes in a classification of V; one of the classes is {O}.
In pirticular, for G = G(2), 2 invertible (the group (--•,t1, 1,2, •} spanned by A), we obtain the 0-A-autothetk sets. For S€ o(G(A)) = c(2, 4') we have' AS = S 26. Central ymetry. The sets H({—1}) = UU — U and H((O, —1}) = U U — U U (0) are complete; c({ —1)) is the family of 0- — lautothetic or 0-
symmetric sets (sets symmetric with respect to 0), e({O, —1)) that of 0-{O, —1)-
endothetic sets: these are the 0-symmetric 0-sets.
If r =2 then all sets are 0-symmetric. If R = {O, 1, —1) then the strong 0-cones coincide with the 0-symmetric 0-sets, and the 0-cones with the 0symmetric sets. In all other cases the sets, 0-sets, flats, 0-flats, 0-cones, strong 0-cones, 0-symmetric sets and 0-symmetric 0-sets in Rare eight distinct families; indeed, the corresponding complete endovector sets intersect differ-
ently the set of four points {O,—1,p,(p,1—p)}, where peR,p*O,1,—1. 27. Nonnegative cases. THEOREM 29. For R = R0 all closed multiplicative subsemigroups C' c (the set of nonnegative numbers) with 1 e C' can be distributed into these 13 mutually disjoint classes:
la. G'=l.
2a. G' nowhere dense, mm (G' — {1)) = p> 1. 3a. mm (G' — {1)) = p > 1, + A c C' (A being the smallest number for which
this holds). 4a.
G' ={A',n = -.-,—1,O,1, •-.;O},A >1.
T. S. MOTZKIN
6a0.
G'
any G' in 2a,3a,4a, the set of the recipn*cals of its elements and of zero.
+1) has 1 as limit point then G' is dense in 14+1; since G' is closed, G' =14+1. If C' fl (14+1) contains an interval then 2 c C' for some 2. If G' * (1), C' n (14 + 1) = (1) then the set of reIndeed, if G' fl
ciprocals of elements of G' — {O} is a closed aemigroup. Finally, if 2€ G',
peG' and log 2/log p is negative and irrational then C' is dense in 14 and thus C' =14. 28. Real cases Tasouu 30. For R = R. all closed multiplicative subsemigroups G' c R with 1 G' can be distributed into these 36 mutually disjoint classes:
The 13 classes of Theorem 29.
lb, ..•,4b'O. For any C' in la, •••,4a'O, the set of its elements and of their negatives. 2c. 3c.
G' nowhere dense, min(G' fl14—{1})=p> 1,max(G'fl , 0, Sd," by holds for inverse stars.
For if v, is a Aoverstar center, 2 > 2., and Vt Se', then v S implies that S contains the segments from v2 to v, — 2,(v — v1), 0 2, 2. Hence v, + p(v1 — = 2(1 — p)/(l + 2p). But o < p < 1, is easily seen to be a 22-overstar center,
for small p.
The proof of the second statement is similar, enlarging v S to a subset of S this time first by consideration of v, and then of v,. 55. The set with 2(v) 2.(2. > 0) are linearly closed in S,I'. holds for inverse stars.
S
of points v €S
The corresponding statement
For if v5 = v1 + p(Vz — v1) e and v, , V2 Sc,", then v S implies that S contains the segments from Vs — 2c,(v — v,) to V5 — 2,(v2 — v3), 0 If p 0 is connected or not. 42.
The strength for convex sets. In {vJ.R0, above or below a point (v, p) + p). Support, n.flat, bounded
+ p) or shall mean in have the usual meaning.
59. If S is a bounded convex set in with more than one point and ifS, consists of all points of above (S, 0) and below every n-flat through an (n — 1)-support of (S, 0) in (RoW, 0) and the opposite parallel support of (S, 1), and Sq of all points above these n-flats and below (S, 1), then S. consists of all points above S, and below Sq.
For the points below S. are those below all the defining segments. Now
'1'. S. MOTZKIN
382
the defining segments through a point (v, 1) of the relative boundary of (S, 1) form the conical part of the relative boundary of the convex set above (S, 0) and below every n-fiat through (v, 1) and a support 8 of (S, 0). Thus the points below Sa are those below all these n-flats, letting v and .3 vary. How-
ever, the points below all n-fiats through a fixed .3 and all (v, 1) are those below the n-flat through .3 and 8', the opposite parallel support of (S, 1). From here the theorem ensues easily.
It follows that the "level sets" on the slope of the relative boundary of The slope of S, S,, or of Sq, determine the relative boundaries of the can be extended and thereby 2(v) defined outside S, with —1 0, then = mm , 2!,.'), but
+ n" if A'÷ These statements follow easily from 2(v', v") max (A(v'), 2(v")). To verify the latter equality consider any minimal segment through (v', v"). It lies in the plane containing the direct product of a minimal segment through v' and a minimal segment through v" and can be replaced by one of these for the purpose of determining 2(v', v"). As in § 43 we prove: THEOREM 70. If the proper ambiconvex set has a center of symmetry s, then = if (1)? § 43, holds; in partiall S, 1 0, z X, and that T is a closed, starshaped
subset of E. If X is complete and T is bounded, then there exists x0 in X such that x0 J(z) and X fl J(x0) = If we assume that X is compact (but not necessarily that T is bounded) then there exists x0 in X such that x0 1(z) and 'p is constant on X fl J(xo). PROOF.
It follows from the semi-continuity of z and ço that J(z) is closed.
now, that X is complete and that T is bounded. In this case, a point x0 in X will satisfy the conclusion oi the lemma if x0 is a maximal element of J(z) n X. We will apply Zorn's lemma to obtain this maximal element; to do this we need only show that any totally ordered subset U of
J(z)nXhas
an
upper bound in X. I' W={z}, we can take x0=z, so we
If xe W,x*z, then 0 z
can assume > 'p(z)
—co.
>
Similar reasoning shows that {'p(x): XE W— (z}} is a mono-
tonic net of real numbers. Since 'p is bounded above on X, this net converges and hence W itself is a p-Cauchy net, that is, given £ > 0 there exists w in
Wsuch that if then p(x—y)<e, i.e., x—yeeT. Now, since T is bounded, given any neighborhood U of the origin in E there exists c > 0 such that cT c U, which shows that W is a Cauchy net in E, and hence converges to an element Yo in the complete set X. To see that Yo
is an upper bound for W, let xe W and suppose ye h,y
x.
Then 0
for each such y; since W converges to Yo, we have (taking all limits over those elements y in W such that y x and using the kp(y — x) ço(y) —
ça(x)
semi-continuity of p and .p) kp(yo—x) urn sup ço(y) —
ço(x)
S°(Yo) — ço(x).
liminfkp(y—x)
Thus, y.
limsup[co(y)—so(x)J =
x, which completes this part of
the proof.
Suppose, now, that X is compact. Then J(z) n X is compact and hence there exists a point x0 6 f(s) n X such that 'p(x0) = sup'p( f(s) fl X). Suppose that xe J(x0) n X. Then xc f(s), so that ço(x) 'p(x.). On the other hand, x e J(x.) implies that ço(x) çc(x0), which hows that 'p is constant on J(x0) n X. 2. Support theorems. In this section we prove our density theorems for support points and support functionals in a locally convex space E. It is possible to define a number of topologies for E but the one which is an
immediate generalization of the norm topology in E5 (when E itself is formed) is the topology of uniform convergence on bounded subsets of E. As is well known (see, e.g., L2, p. 17]) a neighborhood system of the origin for this
topology is given by the polars B° of the members B of the family of all bounded closed convex sets in E which are symmetric about the origin. (For such a set, B° = {f:f€ E5 and f(x) 1 for all x in B}.) Thus, a subset F of E5 is dense in E5 if for each f in E5 and each B in there exists
R. R. PHELFS
g in F such that! — g e B. We now prove two lemmas (analogous to Lemmas 2 and 3 of Lii) which give conditions ubder which two functionals are "close" in this topology. The lemmas themselves require no topological assumptions, so we state and prove them for arbitrary real vector spaces. LEMMA 2.
Suppose that B is a convex symmetric subset of the vector space E,
and suppose that / and g are linear functionals on E with 0 < a = sup! (B) = supg(B) < co. If I g(y)l a whenever 1(y) =0 and y e4aB, then either /— g€(1f2)B° orf+ge(1/2)B°. PROOF. Let h' be the restriction of g to the subspace !_I(O). By hypothesis, if y B n f_*(O), then I h'(y) I 1/4, so, if p is the Minkowski functional for B, then I h' I (i/4)p on I _1(O). By the Hahn-Banach theorem there exists a linear functional h defined on E such that I h I (l/4)p on E and h = h' = g Since g — h vanishes on f1(O), there exists a real number ft such on I E (1/4)B°. We (1/4)p or ±(g — that g — h = ,9f and therefore Ig — must now examine the cases corresponding to the possible values of ft. Note first that if P = 0 then g = h and hence (since sup h(B) 1/4) a 1/4. fore 1,9 aB° C (1/4)B° so that f — 9€ (1/2)B°. We can assume, then, that ft*O. Suppose that ft>O; then either O 0 such that! —
Ag
e B°.
Since k(4a + 1) < a, we can choose x in B such that f(x) > k(4a + 1).
Suppose
that ye 4aB and f(y) =0. Then kp(x ± y) k(1 + 4a) max (1/2, k(4a + 1)); then kp(z) 1/2, and therefore ! + Ag (i/2)B; we conclude that J — Age (i/2)B° c B°.
We now prove a theorem (analogous to Theorem 2 of [1)) which yields, as a corollary, a density theorem for support functionals. TasossE 1. Suppose that C :s a closed convex subset of the complete locally convex space E, and assume that C has nonempty interior. Suppose further that X is a bounded subset of E and that! in E5 is such that supf(C) < inf 1(X). Then if B is any bounded symmetric convex set in E, there exists g in Es and
x0 inC such that supg(C)=g(xo)< infg(X) andf—geB°. PROOF. Let r = sup! (C), o = inff(X) and choose ft such that r
0. Let V = and Choose z inC such that r —f(z) — cony (X u (4.1) ri (x:f(x) let B' be the closure of the bounded symmetric convex set B + cony (V U — V) + [—z, zJ. Let p be the Minkowski functional for B'. If v e V then v e B' and —z B' so that p(v — z) p(v) + p(z) 2. Since X c V c B' we have
o 4 + a' > 0 and by — r)(4 + a1). Lemma 1 there exists x0 in the complete set C such that x0 — z e K(f, k, B') and(K+x0)n C= {x0}. Notethat VcK+x0; indeed, if ye p(v — z) + p(x0 — z)
2 + /C1f (xo — z)
<M +
—1(z)] <M+ k121(j9 — r) =
k1f(v — x0). Since K + x. misses the interior of C the such that 0 < sup h(C) = separation theorem shows the existence of h in suph(B') < oo (since B' is bounded). Since h(x0) = inf h(K + x0) inf h(V) ZM =
k'(fl — r)
k' > 4 + a', Lemma 3 applies to show that there exists A > 0 such that f — ge (B')° c B°, where g = Ah. Finally, to see that inf g(X) > g(x0), note so that v=ô'j9x€V and hence that if x€X, then But then h(x) the proof. h(x0).
ö191h(x0) > h(x0),
so inf h(X) >
h(x0),
which completes
The corâllaries below follow from Theorem 1 in the same way that the analogous corollaries in [1] follow from Theorem 2 of that paper. C is a closed convex set with none,npty interior in the complete locally convex space E, then the support functionals of C are dense (in the topology of uniform convergence on bounded sets) among these functionals in
which are bounded above on C.
CORoLLARY 2. If C is a closed convex set having nonempty interior in the complete locally convex space E, and zf K c E C is a compact convex set, then there exists a functional F in E * which strictly separates K from C and
supports C.
To obtain our theorem on the density of support points, we first state a condition under which our support cone will have nonempty interior. Suppose
that 1€ E and that U is a closed symmetric convex neighborhood of the origin in E such that a supf(U) < oo, Let p be the Minkowski functional for U and suppose k> 0. If there exists x in £ such that kp(x) 0 such that Ax + U c K(f, k, U), and hence K has nonempty interior: We choose A > 0 such that k[Ap(x) + 1] = 21(x) — a. It follows that if ye U then kp(Ax + y) k(Ap(x) + 1) = 21(x) — a f(Ax + y), sincef(y) —a
for all y in U. THEOREM 2 (KLEE).
Suppose
that C is a closed, convex and locally weakly
compact subset of the locally convex space E. dense in the boundary of C. PRooF'.
Then the support points of C are
If z is in the boundary of C, choose a neighborhood V of the origin
such that C' = (z + V) n C is weakly compact. Choose a closed, symmetric convex neighborhood U of the origin such that U + U c V. We first show that z + U contains a support point of C'. To this end, choose y cE C
R. IL PHELPS
398
such that y e z + (112)U and choose (by the separation theorem and local con vexity of E) f in E* such that supf(C) 0) U (viii) If S is closed, convex and has nonempty interior, then 'p is continuous and xeintS} defined by and the interior of S. in the cone (Ax:2 (ix) The function 'p is bounded above on any bounded set.
for all A 0. The Paoop. Observe that ço(x) = if and only if rest of part (i), as well as parts (ii), (iv), and (vii), follows easily from the definitions. (iii) Since S c [1, On the other hand, if 0 < A O such that yeA0U andhencethereexistsanindexa0suchthaty.€2,Uifa a0. IflimsupA, = 00, there
would exist a subnet yp of y, such that
This would mean, then, that assumption that A,*y, = z, eS. Thus, urn sup A. < 00
> A0 and hence A0 U
U, contradicting the since each A0 1, there exists A 1 and a subnet 2, of 2. converging to A. Hence z = A1y, S and therefore yeS', so that S' is closed. r'y; since S is closed, Finally, if ç(x) >0, choose A, > 0 such that 2,, — ç(x) and xc A,,S. Then 4tx çc(xy1x S. so sup (2 0: XC AS) is actually attained. (v) We must show that if ço(x) 0 and x, —' x, then urn sup 'p(x,) ço(x). = we see that x is not in the (closed) Suppose, then, that A > co(x); since a0 and thereAS' if a set AS'. Thus, there exists an index a0 such that fore 'p(x) A if a a0, so that urn sup ço(x) A. Since this is true for each such A, 'p is upper semi-continuous at x. Suppose that —' x in the weak topology. With A as before, we have x€ AS'. Since S is closed and convex, so is AS' and, if E is locally convex, there exists / in E0 such that f(x) < inff(AS') = c. Hence there exists a0 such that 1(x) < c if a a0, so that AS', and the same reasoning as above shows that 'p is weakly upper semicontinuous at x. (vi) Clearly 'p(x + y) 40(x) + 'p(y) if 'p(x) or ço(y) equals —oo or 0. II both are positive and e > 0 choose positive numbers A and 2' such that 2> ApU.
40(x) —
i/2, 2' > ço(y) —
£12
and,
and xc AS, ye A'S. Then (A +
(AS+A'S)cS so that
+ y) €(2 + A'Y' and therefore
(viii) In light of (v), we need only show that 'p is lower semi-continuous at each point x such that x = A0y for some y in the interior of S and A0 > 0
(so 'p(x)> 0). Suppose, then, that x0
x; we must show that 'p(x)
urn mi ç(x).
R. R. PHELPS
400
To do this, it suffices to show that for each r such that 0 < r < 1 there exists Since S is closed, we know that x= if a0 such that the interior of the Since y for some z in S, so that z = 4 and the half-open segment Jz, y) convex set S, we have and Ar 4. Then mt s. Choose A such that 4 < A = is in the interior of S. Since so that
a, such that
if a
e S for a a, and therefore ço(x1)
yJ C < < 1, x there exists rço(x)
— Jco(x)1Aoy,
a,, which completes the proof.
(ix) Suppose that B is a bounded set, and choose a neighborhood Uof the
origin such that rUc U if Id
1 and such that Un S is empty. Choose
A, > 0 such that Bc 20U. If x B, then U if A A. so that x AS if A A,, which shows that ço(x) A, if xeB. If B is a convex subset of a vector space E and if C is a convex subset of
B, we say that a point x in B is an extreme point of B relative to C if x is not interior to any 'nontrivial line segment in B which has one çndpoint in C, that is, if x = Az + (1 — A), (for distinct points z in E,y in C and 0 2 and for the periodic case, R likewise reduces to the least convex solid containing C, under certain auxiliary conditions on C. However, this conjecture turned out to be wrong. There is a basic difference between even and odd dimensions. For odd n, the points of C, in general, lie outside R. For n = 4, for the particular example considered in § 9, the boundary B consists of intersections of pairs of planes (2-flats)
osculating C at two arbitrary points. The author's interest in a (periodic) family of linear inequalities arose in connection with a technical problem relating to the field at a great distance due to varying currents in n antennas. Such a family of inequalities may arise, however, in analysis, as illustrated by the following problem. Consider a polynomial in a = x + iy:
and let us inquire what the conditions are on the (complex) coefficients D1, ..., A, so that the real part of P be positive for a lying on and within the unit circle. Since the harmonic function Re [P(z)J
attains its optima over any region on the boundary of the region, it is sufficient that this condition be applied for j a I = 1. Putting
= Ak — iBk, there results (1.12)
which reduces to
a=
H.
406
(1.13)
A1 cos 0 + B1 sin 8 + ... +
cos nO +
sin nO
—
4,
which is of the form (1.1) in the 2n variables A1, B1, •-•, B. with f1(8) = cos 0, = sin nO, g(0) = —1/2. Thus (1.1) reduces to the condition f2(0) = sin 0, - .
that the Fourier series (1.12) represent a positive harmonic function. Strictly speaking (1.13) differs from (1.1) in that the inequality sign has been replaced by it could, of course, be restored to by changing the signs of both sides of (1.13), thus changing the definitions of , g to their negatives. However, this will not be done in the following, since most of
the descriptions and properties of the admissible region R could be used equally well if the inequality in (1.1) had been reversed ab initio. Similarly, the requirement that + D1c'j(z) + ... + 0 hold over any simply connected region A in the z-plane, where %Oo,SOj, . . ., (1.14)
are
Re [çog(z)
given analytic functions in A, leads to a one-parameter family of in-
equalities of the form (1.1), in the 2n constants A1, B1,. •, A,, B,,, where 0 is a parameter along the boundary of A. 2. Two-dlmenalonal cue. We consider the inequality (1.1) for n = 2: (2.1) x1fj(0) + g(0). Now we may plot (x1 , x2) in a plane. As shown in Figure the inequality
FIGURE 2.1
(2.1) restricts the points to one side of a family of straight lines, one such line resulting for each value of 0. It is evident from Figure 2.1 that the envelope C of the family of straight lines is of interest in describing the admissible region R for the point (x,, x,). To find the equations of the envelope C, we start with equations (1.4), which reduce to (2.2)
Putting
x1f1(O) + x2f2(8)
g(O).
CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 407
0=0=00+40,
(2.3)
there results x1fj(6o) + x2f(Oo) = x1f1(00) + x2f(61) =
(2.4)
+ x,f(02) =
If one obtains the intersection point Pk of the lines 6 = 0,
0=
(k
=
1, 2, .•-) then, letting
limdO—'O, limkdO=6', will, in the limit, approach the envelope C. By subtracting
(2.5)
the points from each equation (2.4) the preceding equation, dividing by 46, and allowing 46 to approach 0, it is evident that the resulting point of intersection P,, will, under the assumption that 1.' g are analytic in 0 (or even that they are of class C'), approach the point P which, at 6 = + 0', satisfies simultaneously the two equations x1f1(0) + x0f0(0) =
(2.6)
= g'(6)
x1f11(8) +
These are linear equations in x1, x,, and if the determinant .0 of the coefficients does not vanish, they admit the unique solution: = F1(6) N1(0)/D(0), x2 = F2(8) = N,(0)/D(o), where N1(0) =
(2.7)
—
= — D(0) =f1(6)f(0) As 0 varies, this solution describes the envelope C. Summarizing, the envelope C is given by N1(8)
C: x1 = FL(0),
(2.8)
x2 = F2(6),
where F1, F2 are as in equation (2.7), and is finite provided D(6)
(2.9)
—f*(0)f(6) * 0.
Of interest in describing the relation of R and C is the concavity of C and its curvature, defined by (2.10) k = dço/ds, where ço is the angle that the tangent to C in the direction of increasing 0 makes with the positive xt-axis: ço
(1=
d/dO),
(2.12)
=
= tan1F2'/F11,
and ds is the length element oil C, defined by
ds=vd6,
H. PORITSKY
408
The angle ço may also be defined as = (2.13) with the x1-axis. since the normal to the line (2.2) makes an angle ± It will be seen from Figure 2.1 that if at 8 = 8' the envelope C is concave to the admissible side of (2.1), the portion of C near 8' forms part of the boundary B of R, unless as 0 varies sufficiently to either side, this portion of C is "cut off" by lying on the inadmissible side of member tines of (2.1). We shall now indicate some of the possible relations between R and its boundary B, and the curve C, but no exhaustive enumeration of all the possible cases will be made. Suppose first that 0 is confined to a finite interval (1.2), and suppose that the envelope C for that interval is finite, always concave toward the admissible side of the straight lines, and such that the total rotation of C, 4co =
(2.14)
ds =
= tan_hftlfl]0
is numerically less than it. Then, as shown on Figure 2.1, the point (x1, x1)
is confined to the region R bounded by the envelope C, and halves of the terminal lines (i.e., corresponding to the end points 0 = 0., 0 = of the
interval), proceeding away from C. If the same conditions regarding the envelope C hold, but the total rotation of the envelope lies between ir and 2ir, then R is bounded by C and finite segments on the terminal lines, as shown in Figure 2.2.
FIGURE 2.2
g(8) in (2.1) are periodic, of the same period (see Then the envelope C, if finite, is a closed curve. If C is equations (1.3)). always concave toward the admissible side and is a simple ctirve (i.e., if it does not intersect itself), then the region R consists of the interior of C, as
Suppose next that
shown in Figure 2.3. In Figures. 2.1, 2.2, 2.3, the condition regarding concavity
can be replaced by the statements: the curve C is always concave in the same direction, and one point P1 of C lies on the admissible side of the tangent to
Cat some other point P2. Stating the result illustrated in Figure 2.3 more formally, we have the following: THEoREM 1.
Let It, f. g in (2.1) be of class C";
suppose that they are
CONVEX SPACES ASSOCIATED WITH A FAMILY OF LINEAR INEQUALITIES 409
FIGURE 2.3
periodic in 6, with the same (smallest) period ii, and are such that
(F1')' + (F,') * 0,
(2.15)
(2.16)
D=
* 0,
—
(2.17)
=±2r,
(2.18) (2.19)
[F1(61)f1(6,) + F,(61)f,(O,) — 9(02)1
O, O 0 with M(H,
e) = 0; S and a sequence of
(2) there exists a sequence of distinct elements such
that (SI,
.
•, s,,}
c W%.
Suppose that (2) is fulfilled. Let H be the set consisting Suppose that A M(H, e) for some Since N(A) is finite and N(A) c H, we have N(A) c {s1, . . ., for some
PROOF.
of all elements of the sequence e
< 1.
n so that N(A) c
It follows that 1 =
A(N(2))
A(
0. By (1.1), there exists a finite nonvoid K1 such that = 0. Since K1 is finite, we have H — K1 * 0 and it follows from (1.1) that £14) = 0 for some nonvoid finite K2 c H — K1. M(H — K1, ii Take now H — K1 — K2 * 0 and repeat the process. In this manner we obtain a sequence of pairwise disjoint nonvoid finite sets K1, K2, ... such thats * 0 for each n. Let P,, be the subset of those fl x [s,, x K,, for which there exists a We with '(S., W. •
It follows that each P,, is nonvoid and that the
I',,
fulfill the "restriction"
property. By Lemma (1.2) we obtain a sequence K1 such that Es1, . . ., s,,] P,. for each n. The sets being mutually disjoint, the sequence is distinct.
In the sequel we shall be frequently dealing with the following situation: We are given two sets S and A and a subset W of S x A. For each aeA, let W(a) be the set of those s €S for which Es, a] W; similarly, if s€ S, we denote by W(s) the set of those a A for which Es, a] £ W. The family will consist of all sets W(a) with a e A. In this situation, the condition for the existence of convex means may be reformulated as follows: (1.4)
COROLLARY.
The following conditions are equivalent:
(1) there exists an infinite HcSandan e>0 with (2) there exists a sequence of distinct elements S,, S and a sequence a,, such that '(Si, ... s,,} c W(a,.) for each n; (3) there exists a sequence of distinct elements s,, S and a sequence a,,
A A
such that W contains the "triangle" consisting of points [se, a,,1 for i it; (4) there exists a sequence of distinct elements s,, e S and a sequence a,, A such that s,,E W(a,,) fl W(a,+1) fl •.. for each n;
VLASTIMIL P'rAK
440
(5) there exists a sequence of distinct elements
S such that W(s,) fl
A
is nonvoid for each n. PROOF.
Immediate.
2. An illustration. In this section we intend to show how the theorem on convex means can be applied. We shall apply it to the proof of a classical theorem; although the result we are going to prove is by far not the best possible, the proof represents a typical application of the lemma on convex means and puts into evidence the idea underlying all further applications.
Let T be compact Hausdorff, Xn C( T) and 1. Suppose that — 0 for each t T. Let e > 0 be given. Then there exist non-negative numbers A,,-• -, with = 1 such that < (2.1)
I
lim
PROOF. Let S be the set of all natural numbers and define a set W c S x T in the following manner: [s, tI W if and only if x,(t) e. Let be the family of all W(t) for t T. Let us show first that our problem will be solved if we find a A eM(S, e). Indeed, we intend to show that To see that, take an arbitrary t T. We A(s)x,1 0, the set M(S, s) is empty. Then there exist two sequences S and A suck that
M(a,) fl ... (spice s,, ..
s,,
and
A
M(a,,_,) A E
A W(an+j) A
the sequence
is distinct).
A COMBINATORIAL LEMMA ON THE EXISTENCE OF' CONVEX MEANS
Since M(S,
PROOF.
that M(S,
e12)
€)
= 0.
0,
441
by (1.1) there exists a nonvoid finite K1 such
Let P1 be the set of those s
K1
for which
W(s) * 0. Clearly P1 0. For each p e P1 , p = s, choose an a(p) e W(s) so that s€ W(a(p)). Put M1 = fl M(a(p)) for p E P1 so that M1 * 0 by our assumption. By (1.1), there exists a nonvoid finite K2c M1 with M(MI, e14) = 0. Let P2 be the set of those pairs [s1 , s2J e K1 x K2 for which W(s1) fl W(s2) 0. Clearly P2 * 0. For each p e P2 , p = Es2 , s,], choose an a(p) W(s1) fl W(s2) so that s1 , s2 W(a(p)). Put M2 = fl M(a(p)) for p £ P1 U P2 so that M± * 0. Suppose we have already defined the sets K1, •, K,, and P1, •, P,,-1 and a mapping a(p) on Pi U U P,,-1 to A such that M(M,,_1, (1f2")e) = 0 where M,,_1 — fl W(a(p)) for fl ••- fl pEPI U U P,,-1. Let P,, be the set of all [SI, x .•. x K,, for which W(s1) fl ... Ii W(s,,) *0. Clearly I',, * 0. For each P€P,,,p = [s1,.. - -
chooseana(p)eW(s1)fl ••-fl W(s,,). PutM,,=flM(a(p)) forpEP1U-.-LJP,,, so that M,, * 0. By (1.1) there exists a nonvoid finite K,,+1 c M,, with M(M,,, Ii fl = 0. We have thus obtained a sequence of nonvoid finite sets K,, c M,,_1; the sets P,, satisfy the "restriction" property of Lemma (1.2). Hence there exists a sequence s, such that = [s1, -, s,,] e P,, for each n. Put a,, a(p,,). It follows that {s,, •, s,,} W(a,,). Further, s,, K,, c M,,..1 c M(a,) fl - - - fl M(a,,1). The proof is complete. -
-
We are now able to prove a general result which includes both the Eberlein theorem and the Krein theorem. We begin with a definition.
(3.2) DEFINITION. Let E be a convex space. A set A c E is said to fulfill the double limit condition if it is impossible to find a neighborhood of zero U in E and two sequences a1 e A and 4 U° such that lim, and
urn1 lim. 0 and each subsequence < e; such that exists for each n we have (4) for each sequence h, H such that urn5 urn urn = 0; I
I
3
converges to zero almost uniformly on H; converges to zero almost uniformly on T.
(5) (6) PROOF.
Assume (1), take an infinite set R of natural numbers and a positive
< t}. These sets are open in T each r R, let U(r) = T; I and form a covering of Tso that there exists a finite Kc R with = €.
For
I
It follows that the convergence is almost uniform on T and we have (6) which, in its turn, implies (5). We have shown in the preceding (5) implies (4). If (4) is satisfied and an infinite R and a positive given, it T.
follows from the preceding lemma that there exists a finite Fc R and a
such that < e for each he H. Since H is dense in T, we have so that (3) is proved. Let (3) be fuifihled and suppose there exists a functional x' C( T)' of norm one and a positive £ with for each n in an infinite index set R. , x'> Take a convex mean y = with Fc R such that lyl < €/2. We have t x' < e/2 which is a contradiction. <x,, x'> = 0 such that for each xeE there exists a yeM with A bounded set M c E' is (5.6) LEMMA. Let E be a normed linear space. E)-closed absolutely convex extension norm-generating if and only if its contains a multiple of the unit cell of E'. PRoOF. Let M be norm-generating and suppose that xeM°. We have
aix I sup (x, M> 1 so that x e (1/a)U where U is the unit cell of E. Hence c (1/a)U whence aU° c M°°. On the other hand, if aU° c M°°, we have sup(x,M> sup(x,M°°> sup<x,aU°> = alxt. (5.7)
THEOREM.
Let E be a normed linear space and
a sequcnce in E.
Then the following conditions are equivalent: converges weakly to zero; (1) (2) x,. is bounded and converges to zero on a norm-generating subset of E' which is a(E', E)-com pact;
is bounded and converges to zero almost uniformly on some norm(3) generating subset of E'; (4) x% is bounded and there exists a norm-generating subset A c E' such that urn urn <xx, a> = 0 n
j
for each sequence aj e A such that the urn, <xx, a'> exists for each n. converges to zero in the weak topology of E. Then PROOF. Suppose that is bounded and converges to zero almost uniformly on any a(E', E)-compact
subset of E'. Now any norm-generating subset of E' is bounded in the
norm and hence contained in a c(E', E)-compact set. Suppose now that (4) is satisfied. Since A is norm-generating, there exist positive numbers a and such that sup<x,A>
for each xe E. It follows that the mapping T which assigns to each xe E the function Tx ye B(A) defined by y(a) = (x, a> is an algebraic and topo= converge logical isomorphism. Since (4) is fulfilled, it follows that to zero in the weak topology of B(A) by (5.3). If fe E', there exists a g e B(A)' such that <TI,9> = for each x e E so that = converges to zero. (5.8)
Let M be the Banach space of all bounded measurable functions on
(0, 1> modulo functions zero almost everywhere with the norm I x Let
I
ess sup x(t) I.
A be the set of all functions of the form (1/p(B))cfi, where
characteristic function of a measurable set B. Let
is the
be a bounded sequence in
A COMBINATORJAL LEMMA ON THE EXiSTENCE OF CONVEX MEANS
M.
447
Then x,, converges weakly to zero if and only f =
Jim Jim
0
xN(t)aJ(t)dt exists for each n. for each sequence a A such that It is easy to see that the elements of A constitute a norm-generating
subset of M'. be a measure space with p such that be a sequence of sets in
(5.9) Let (1',
p)
0 and p(T) < co and let a > 0 for each n. Then
there exists a subsequence n1 < n2 < ••- such that AN1 (1 void for each k.
n
fl
is non-
PROOF. Let S be the set of natural numbers, and let Wc S x T be defined as follows: (s,t]e Wif and only if tEA,. Let /1 be the family of all W(t) 0, we can with t T. If we show that M(S, , e) is empty for some use (1.4) to show that there is an increasing sequence < s2 < ... such that
is nonempty for each k. Since obviously W(s) = A., the W(s1) fl fl theorem will be proved. To show that M(S, Y/', e) is empty, suppose there is a AG M(S, Consider the linear subspace E of B(T) spanned by the of A,,. characteristic functions It is easy to see that it is possible to define a bounded linear functional f on E by the formula
<Eaixs,f> = Since AeM(S,7/',€), we have 0
e
for each I T. It follows that = ep(T)
On the other hand A(s)p(A,) a
so that we obtain a contradiction if ep(T) < a. 6. Weak colnpactues8. A slight technical refinement of the preceding methods yields characterizations of weak compactness in linear spaces. (6.1) DEFINITIONS. If S is a completely regular topological space, let Ce(S) be the Banach space of all bounded continuous functions on S with the norm 'Cs
If E Ce(S), let be the unit ball of E' in the topology o(E', E) so that S may be considered as a topological subspace of S. if P and Q are two topological spaces, a function h(p, q) on P x Q is said to be separately continuous if it is a continuous function of each variable, the other variable is
VLASTIMIL PTAK
448
kept fixed. We shall say that h satisfies the double limit condition if it is impossible to find two sequences p P and q e Q such that the limits lim, other.
,
qi)
q,) both exist and are different from each
and lim,
All results of this section are based on the following proposition. Let T be a completely regular topological space and let A be a bounded T) such that the double limit condition is satisfied on A x T. subset of E Then A is weakly relatively compact in a(E, E'). (6.2)
PRooF. Consider A imbedded in a cartesian product P of real lines, one for each point t e T. Since A is bounded in E, the closure of A in P is compact. Take any r in this closure. Our theorem will be proved if we A show that, for each £ > 0, there exists a convex mean b = all 1 e T. I £ for Indeed, this shows first that r(t) is such that 1 r(t) — b(t) continuous on T and, further, that it also belongs to the a(E, E') closure of A in E.
To prove the possibility of approximation, take a fixed e > 0 and consider in
A x T the sets
M={[a,tj€A x T;II to S x T by putting = for each y€ If s€S is fixed, the function <s,y> = is Continuous on T by definition of the topology on T. Consider now, in F', the linear space L(T) algebraically generated by the evaluation functionals I e T. Let W be the closure of h(S) in F in the topology a(F, F'). Since W is compact in this topology and the topology ci(F, L(T)) is coarser, these two topologies coincide on W. Now if s0 —' s0 then h(s0) in o(F, L(T)) and consequently, in o(F, F') as well. It follows that <S0 ,Yo> =
=
for each Yo e 1' so that <s, y> is separately continuous on S x 1'. By Lemma (6.3) the duality extended to S x T again fulfills the double limit condition and is clearly bounded. We complete the proof by observing that the same construction may now be repeated with T and S instead of S and T.
It is easy to see that this general theorem contains as a particular case the Eberlein-Krein theorem (3.3). The proof is immediate and is left to the reader. The present Theorem (6.5) enables us to weaken the assumptions since now we need not assdme the double limit condition on the whole of A x U°. It is sufficient to have the double limit condition on A x T only, T being
some norm-generating subset of U°. A number of similar results may be obtained, the formulation and proofs of which are left to the reader. Let us conclude by listing some conditions for weak compactness in spaces of continuous functions. (6.6)
DEFINITION.
Let T be a compact Hausdorif space and A c C(T). The
family A is said to be quasi-equicontinuous if the following condition is satisfied: given a directed set t0 — t0 in T then for each a0 and e > 0 there such that exists a finite set K of indices mm
I a(t0) — a(t0) I
0. (Here (x, y) —s <x, y> denotes, as usual, the canonical bilinear form on E x F.) When C is a saturated family of weakly bounded subsets of F whose union is F, it is well known that the polars of the members of C form a neighborhood basis of 0 for a locally convex topology on E, called is consistent with <E, F> if F, identified with a subthe C-topology space of the algebraical dual E* of E, is the topological dual of E for in
I
The duality between normal cones and C-cones is established by the following
theorem which, in somewhat more general form, was proved in [6, 1] (see also [7, § 6]). THEOREM A. Let <E, F> be a dual system, C a family of weakly bounded subsets of F such that the C-topology on E is consistent with <E. F>. A cone K in E is normal for the C-topology if and only if K' is a strict C-cone in F. In particular, K is weakly normal in E if and only if F = K' — K'.
Assume that E, F are locally convex spaces over the same scalar field, and denote by 2'(E, F) the vector space of all continuous linear maps on E into F. Equipped with the topology of uniform convergence on a family C (as above) of subsets of E,. we shall denote this space by 22€(E, F). If K and H are cones in E and F, respectively, the set of all Te .9'(E, F), for which F); conditions under which this cone is proper TK c H, forms a cone in F) were given in [7, § 8]. We shall need the following and closed in result whose simple proof may be found in 17, (8.3)]. THEOREM B.
If K is an C.cone in E and H is normal in F, then
.
=
F). fT 9'(E, F): TK c H) is a normal cone in A third result needed later in this paper is a theorem [7, (7.2)] on the convergence of section filters of a directed set In an ordered locally convex space. Let E be an ordered locally convex space with positive cone K, and let M be a (nonempty) subset of E directed for ". The family of sections M. = (y e M: y x} forms a filter base in E; the corresponding filter is called the filter of sections of M and denoted by THEOREM C.
Let E be an ordered locally convex space with normal Positive
H. H. SCHAEFER
454
cone K, and let M be a nonempty directed subset of E. The weak convergence implies its convergence for the given topology on E. of
The remainder of this section is concerned with the definition of several concepts related to the spectrum of an element in a locally convex algebra. A locally convex algebra is a locally convex space and an algebra such that multiplication is separately continuous; unless the contrary is explicity stated, we assume such an algebra A to be defined over the complex field C, and we shall always assume that A has a unit element e. A locally convex algebra is ordered if its underlying locally convex space is ordered with a weakly normal positive cone K such that K contains e and the product of any commuting pair of its elements. An example of an ordered locally convex algebra is furnished by the algebra of continuous endomorphisms of Hilbert space (with K the cone of positive Hermitian operators), under the topology of either bounded or pointwise convergence; other examples are the algebra of space and, under certain condicontinuous compjex functions on a tions (cf. Theorem B), the algebra of continuous endomorphisms of an ordered locally convex space (under a suitable s-topology and the induced order). If A is a locally convex algebra and a e A, the spectrum o(a) of a in the one-point compactification of the complex plane (the Riemann sphere) is the complement of the largest open set in which A —* (Ae — ay* is locally holomorphic; this function, unless its domain is empty, is called the resolvent of a. the algebra of continuous, Let X be a compact (Hausdorif) space, complex-valued functions on X under the uniform topology, A a locally convex algebra. A spectral measure (on X into A) is a continuous homomorphism of the algebra ct'(X), with values in A; the supPort of- a spectral measure p
is the complement in X of the• largest open set U such that p(f) = 0 for every fe whose support is contained in U. A spectral algebra is the range of a spectral measure, and a spectral element of a locally convex alge-
bra A is an element contained in a spectral subalgebra of A. When A is a the (real) Banach algebra of all locally ctnvex algebra over R; and continuous real functions on X, one defines in a completely analogous manner
the notions of real spectral measure, real spectral algebra, and real spectral element. PART I.
GENERAL PosiTivE ELEMENTS
1. The generalized Pringeheim theorem. The theorem that we shall prove first was established, under restriction to Banach spaces, in [11]. The proof in made use of the classical Pringsheim theorem3 which asserts that a
power series with non-negative coefficients and radius of convergence 1 defines
an analytic function I such that the element of 1, represented by the power series in question, is singular at z = 1: To this, the result can be added that = 1; both assertions if this singularity is a pole, its order is maximal on persist as long as the coefficients of the series remain in a sector, vertex I
Cf. Landau, Dar8tellung und Begrundung einiger neuerer Ergebniue der tionentheorie, Springer, Berlin, 1929; §17.
CONVEX CONES AND SPECTRAL THEORY
455
at 0, of central angle [(t — 0
0
(t —
p)'
(p
k. Because K is a weakly normal cone, this implies, for any
p>k,that
a,t' cos nO and
(1 — 0
converge to 0 for o(E, E') as t —
(1 —
t)'
aj' sin nO 0
1.
Thus if C is a pole of I of order m, it
H. H. SCHAEFER
456
follows that m
k and the theorem is proved.
COROLLARY 1. Let A be an ordered, semi-complete locally convex algebra and r(a) c(a). The spectral radius r(a) is in a(a), and let a be positive with is a pole of the resolvent of a, it is of maximum order on (2 I r(a). PROOF. Since c,5 is not in the spectrum c(a) of a, the resolvent R(A) of a exists in a neighborhood of that point, and its expansion at 00 is necessarily of the form
(00 = e)
R(A) =
and f(z) = R('A). If r(a) = 0, then f is an entire function and the Set z = first part of the corollary follows from Liouville's theorem while the second Theorem 1 becomes applicable after a simple becomes trivial. If r(a) > normalization, since by the definition of an ordered locally convex algebra, all coefficients of I are contained in a weakly' normal cone of A. The preceding corollary to
a
of
2'(E). This, in turn, occurs
T€2'(E)
invariant
a cone K in E that satisfies certain assumptions. When E is a Banach space,
it was first shown in [2] that TKc K implies r(T)€c(T) if K and K' are both normal for the respective norm topologies of E and E'. We show that TK c K implies the assertions of Corollary 1 when K and K' are normal and weakly normal, respectively; the spectrum of T is understood with respect to the topology of simple convergence for which topology we to be semi-complete. When E is a Banach space, this notion of spectrum coincides with the usual one. assume
COROLLARY 2. Let E be an ordered locally convex space whose Positive cone is normal and generating,' and let be semi-complete for the topology of simple convergence. The assertions of Corollary 1 apply to every continuous, Positive endoinor/ihism T of E such that co a( T). PROOF. On account of the preceding remarks, we have only to show that for its induced order and the topology of simple convergence, is an ordered locally convex algebra. But it is clear that the positive cone in is closed, and that it contains the identity map and the product of
any commuting pair (in fact, of any pair) of its elements; it is normal by Theorem B and hence weakly normal (Theorem A). Perhaps the best known result on spectral properties of positive operators
is the theorem that (under suitable, but very general assumptions) every positive compact endomo(phism of an ordered Banach space with positive spectral radius rCT.) as a characteristic number with (at least) one positive characteristic vector. This theorem, which has a comparatively long history, was first proved in the stated generality in [5]. A more general A cone K is generating in E if E K — K. By Theorem A, this property is equivalent with the weak normality of K'.
CONVEX CONES AM) SPECTRAL THEORY
457
theorem—but still in the framework of Banach spaces—with what might be called a geometric proof, appeared in [31 and was extended to locally convex spaces in [71. We are going to give a new proof of an extended version of the Krein-Rutman theorem. It is surprising that no normality condition has to be imposed in Theorem 2 (see below); but this is only apparent and the Krein-Rutman theorem is in fact a corollary of the generalized Pringsheim theorem, if not an obvious one. THEOREM 2.
Let E be an ordered, semi.complete locally convex space whose
positive cone is total in K. If T is a Positive, compact endomorphism of E with r(T) > 0, then r(T) is a characteristic number of T with (at least) one positive characteristic vector. Furthermore, r( T) is a pole of the resolveni of maximal order on
I
Al
= r(T).
The essential part of the proof consists in showing that r(T)Ea(T). is equipped with the topology of simple convergence.) Since the positive cone K of K is closed and proper, it follows from routine considerations (or from [6, (1.7))) that its dual K' is total in K' for a(E', K); hence if we let = K' — K', <E, is a nondegenerate dual system and K is normal for a(E, by Theorem A. TK c K implies that T is continuous for this latter topology. Denote by and respectively, the spaces of endomorphisms of E continuous for o(E, K') and o(E, E), each equipped with the topology of simple convergence on E and K, respectively. Since T"Kc K for all natural numbers n, it follows that PROOF.
(*) is
the expansion, valid for I Al> r(T), of R(A) both as a member of 2'(E)
This implies, in particular, that r0(T) r(T) if r0(T) is the spectral radius of T as a member of (the completion of) On the other hand, Theorem B implies that the cone C whose elements leave K invariant, is normal in thus by Theorem 1, r0(T) is in the spectrum of T€ .5f0(E). On the other hand, since T is compact (for the given topology of E),s there exists a number E, I El = r(T), which is a characteristic and
.
value of T; this implies r(T) = r0(T). Let us show that r(T) cannot be regular
for the resolvent of Te &'(E). It follows from (*) that R(A)K c K for all A> r; if r were regular we would have R(r)K c K since K is closed, hence R(r) e . WO and the expansion of R(A) e in a neighborhood of A = would be R(A) = R(r)[I + (r — A)R(r) + (r — A)2R(r)2
+ ...];
since the series converges to an element of it is clear that the r would in fact imply that R(A), as a member of is holomorphic at A r(T) = r0(T). Since this is impossible, we have shown that r(T) is in the spectrum of Te Since T iscompact, r is a pole of R(A) in .Y'(E) and, therefore, in it follows convergence of (**) at any point A
7' is compact if 7'U is relatively compact in E for a suitable neighborhood U of 0.
H. H. SCHAEFER
from Theorem I that, when A R(A) is considered as taking its values in the this pole is of maximal order on (A = r; clearly then completion of maximal among the poles, on (A = r, of the resolvent of T the order of r is I
I
in
to be shown that there exists at least one characteristic vector of T belonging to r(T). But if P is the leading coefficient in the x0 K principal part of R(A) at A = r, every nonzero element in the range of P is a characteristic vector of T with respect to r; on the other hand, since It
R(A)Kc K for all A > r and
P = lim (A — r)kR(A)
k is the order of the pole r), one concludes that PK c K. Thus, K being a total subset of E, there exists Yo K such that x0 = Py0 is a (nonzero) characteristic vector of T in K. (where
CoRoLLARY.
Under the assumptzons of Theorem 2, r(T) is also a characteristic
value of the adjoint T' of T in E', with (at least) one characteristic vector in the dual K' of the positive cone K of E. PRoOF. It is easily verified that T' is a compact endomorphism of E' for E) of uniform convergence on the compact convex subsets the topology Since this topology is consistent with <E, E'>, it follows that K' is a of E. closed, proper, total cone in E' with respect to ic(E', E). The observation that T'(K') c K' and that r(T) = r(T') concludes the proof. 2. Quasi-interior positive operators. While Theorems 1 and 2 of the previous section appear to reflect the essence of what could be expected of positive elements, or compact positive endomorphisms, with respect to their spectral behaviour at A = r, it is known from the finite dimensional case (ct. Introduction) that for certain types of positive endomorphisms (e.g., matrices with strictly positive entries), considerably stronger conclusions hold. Most of these additional results on the spectral radius r( T) could be established in [51 for "strongly positive" compact endomorphisms of an ordered Banach
space; here T is strongly positive if for each nonzero x€K, there exists n =n(x) such that T'x is interior to K. Thus this definition excludes, a
priori, many of the concrete naturally ordered Banach spaces (such as Hilbert from the discussion; moreover, in a non-formable space, no weakly space normal cone can have interior points [7, (7.6)]. In an unpublished paper by R. E. Fullerton,6 and independently in [!O], a weaker concept was introduced which covers most cases in which an intuitive notion of "strictly positive" elements is present: An element x of the positive cone K of an ordered topological vector space E is quasi-interior to K if K fl (x — K) is a total
subset of E. The relation of this concept to those of "support point" of K and (in the case of a vector lattice) of "weak order unit," has been discussed 6 Quasi-interior points of C0fl88 in a linear space, University of Maryland, 1957. In this paper, the basic properties of quasi-interior points are developed but no applications to linear mappings are made.
CONVEX CONES AND SPECTRAL THEORY
459
in some detail in [8]. With the aid of this notion, one defines [10] a positive endomorphism T of E to be a quasi-interior map if 00 c(T), and if there exists p > r(T) such that
y = R(p)Tx= is quasi-interior to K for every nonzero x E K. Since, when K has nonempty
interior, every quasi-interior point of K is an interior point and conversely, the present notion is weaker than that of strong positivity. A typical example of a quasi-interior map on 4 (in its natural order) is given by a bounded such that ti,k 0 and such that for each matrix T k) of subscripts, there exists n with > 0. As the principal available result on quasi-interior positive maps we quote the following theorem whose proof, which can be carried over verbally from [10, Theorem 2), will be omitted. Recall that a linear form f on (an ordered vector space) E is strictly positive if Re /(x) > 0 for every nonzero x eK. A locally convex space is called a locally convex vector lattice if it is lattice ordered such that its positive cone is normal and the lattice operations are continuous. THEOREM 3.
Let E be an ordered, semi-complete locally convex space, T a
quasi-interior Positive endomorphism. If the spectral radius r is a Pole of the
resolvent of T, then:
(1) r> 0 and r is a simple Pole of the resolvent. (2) Every characteristic vector Pertaining to r, of T in K (of the adjoint T' in K'), is quasi-interior to K (is a strictly positive linear form). (3) Each of the following assumptions implies that the dimension d(r) of the nullspace of (rI — T) is one:
(a) K has nonempty interior; (b) d(r) is finite; (c) E is a locally convex vector lattice. 3. A Tauberian theorem. Problem8. S. Karlin [4] has applied several theo-
rems on power series with non-negative coefficients to the resolvent of a positive operator in an ordered Banach space. Since the method Qf proof applies immediately to the case of an ordered locally convex algebra, we wish to formulate, as an example, the Tauberian theorem proved in [4] under the present more general conditions. THEOREM 4. Let a be a Positive element in an ordered, semi-complete locally convex algebra such that r(a) = 1 and r(a) is a Pole of the resolvent of order k. If e0 is the residue of the resolvent R(A; a) at A 1, then (weakly)
urn
=
(a
—
e denotes, as always, the unit element of the algebra A.) The proof of this theorem is, by Theorem A, immediate from the fact that the leading
coefficient of the principal part in the expansion of R(A; a) at A = 1 is (a —
H. H. SCHAEFER
460
and from the classical result' that a,
a
whenever a, 0 are the coefficients of a power series with radius of convergence 1 and representating a scalar function f which has a pole of order
k (k 0) at 1, with a the leading coefficient in the principal part of 1. Application of Theorem 4 to quasi-interior maps yields the following ergodic theorem. COROLLARY. Let E be an ordered, semi-complete locally convex space whose Positive cone is weakly normal and generating. Let T be a quasi-interior Positive map such that r(T) = 1 is a pole of R(A; T), and denote by P the residue of R(A; T) at 1. Then one has Jim
E = 0 be a number such that A —* R(A) = (Ae — is holomorphic in IA— A01 < €; let U= {t€X: 11(t) — A01 <e/2} and let ge be a function whose support is contained in U. For each A satisfying by setting I A — A0 I > 3e/4, we define an element g(2) e 0
It is assumed that t'(l)
e.
If p(1)
e then c(a) = fiXo) U 101.
CONVEX CONES AND SPECTRAL THEORY
g(A;
t) =
if
A
1(1) * A
iff(t)=A.
0
It is readily verified that A — g(A) is holomorphic in P = (A: A — A0 I > 3e/4}, in the annulus and so is A 12[g(A)). We wish to show that ii[q(2)] 3e/4 < A — A0 < e; but obviously, the latter identity is valid in a neighborhood of infinity, hence (Ae — a)j4g(A)] = p(q) in that neighborhood. This implies, in by the identity theorem for analytic functions, that (Ae — a)u[g(A)] = D and, therefore, the assertion. Since 1 R(A) is holomorphic in A — A0 I < s, I
has a holomorphic extension to all of the complex plane; from
A —o
g(A)
= 0 throughout. proved.
p[g(A)J
= 0 and hence, by Liouville's theorem, = 0, the theorem is Since this clearly implies
= 0 we obtain limA...
It follows from Theorem 3 that a real spectral element of a locally convex is algebra is a spectral element with rea.l spectrum; conversely, if a = an element with real spectrum, then f must be real valued on the support of A spectral element is Positive if its spectrum is contained in the nonnegative real numbers; such an element is clearly positive for every ordering of A for which any presenting spectral measure (i.e., a spectral measure such that a = p(f)) is positive. It is evident that each spectral element a can be written as a = a1 + La2 where a1 , a± are real spectral elements such that the product (Theorem 2) of their associated real spectral measures exists; this condition, as may be concluded from Theorem 2, is also sufficient for a1 + ia1 to be spectral; furthermore, it makes the representation unique [121. Similarly, every real spectral element b = pCI) has a (minimal) decomposition b = b1 — b where b1 = and b2 = are positive. The following theorem
is concerned with the existence of positive spectral elements in a semicomplete locally convex algebra A. THEOREM 4. For a A to be a positive,spectral element, it is necessary and sufficient that there exist an ordering of A and a constant r > 0 such that
o
a
ye.
PROOF. If a = p(f) is a positive spectral element of A and if denotes the restriction of f to the support X0 of then it follows that 0 (Theorem 3). By the corollary of Theorem 1, there exists an ordering of A for which p is positive. Since it is-clear that 0 a iifo lie for any such order, the condition is necessary. To prove its sufficiency we assume, without loss of generality, that 0 a e for some order structure of A. We construct a real spectral measure presenting a as follows. Denote by P,, the polynomial on [0, 1]
It is well known that these polynomials (which are the Bernstein polynomials
to within a positive factor) permit the representation of any non•negative polynomial P as a linear combination a,,,,P,,,,, with a,,,, 0. Thus if we
H. H. SCHAEFER
466
define a linear mapping p by = a"'(e
—
homomorphism of the (real) algebra .9 of real polynomials on [0, 11 into A which maps the positive cone of .9 into the positive cone of A. This, as we have observed on earlier occasions, implies a unique, continuous extension fi to the continuity of p; thus which is a real spectral measure such that a = ft(1). It follows from Theorem 3 that 0(a) c [0, 11 and the proof is complete. a
COROLLARY 1.
tion c1e
a
In any ordered, semi-complete locally convex algebra, the rela-
c2e (c1 , c2 e R) implies g(a) c [c1, c2].
The essential contents of Theorem 4 can be transformed into a necessary and sufficient condition for an a £ A to be a real spectral element. Let at(r) = — a(r eR), and denote by C(a; r) the (convex) conical extenre + a, a0(r) = sion of (i.e., the smallest cone containing) the set CoRoi.LAU1Y 2.
m, fl C N) U For a to be a real spectral element of A, it is necessary and
sufficient that for at least one constant r >
0,
C(a; r) is a weakly normal cone
is
weakly normal. If K denotes
mA.
PROOF.
Let To > 0 be such that C(a;
To)
the closure C(a; To) in A, then K is weakly normal, an abelian subset of A, and contains a1(r0)° = e. Hence K is the positive cone for an ordering of A such that —Toe a roe. This implies 0 a + roe 2y0e whence it follows
by Theorem 4 that a + roe is a positive and therefore a a real spectral element of A. The necessity of the condition is easily proved with the aid of Theorem 1. A corresponding condition holds for arbitrary (complex) spectral elements: For c A to be a complex spectral element, it is necessary and sufficient that
there exist a representation c = a + ib where a, b commute and such that for some r > 0, C(a; r)C(b; r) is contained in a weakly normal cone. The necessity
of this condition follows from a remark made earlier in this section; the proof of sufficiency proceeds by constructing two spectral measures presenting a, b, respectively, and presenting c by their product (Theorem 2). 3. Spectral algebras. As the preceding section shows, spectral elements of a locally convex algebra can be completely described in terms of convex
cones (or equivalently, order relations) in A; we proceed to show that a similar situation prevails with respect to the spectral subalgebras of A in which spectral elements (by their definition) are imbedded. Our first theorem concerns the topological properties of spectral subalgebras.
Two locally convex algebras are called equivalent if there exists a topo. logical and algebraic isomorphism of one onto the other; we denote, as before, the spectral radius of a e A by r(a). THEOREM 5. Let A be a spectral subalgebra of A in its induced topology.
CONVEX CONES AND SPECTRAL THEORY
Consider the following properties: (a) a —' r(a) is continuous on A;
(b) A is equivalent to an algebra (c) A is closed in A. Then (a) is equivalent to (b), and (a) implies (c). PROOF. (a) (b). Let A be the range of a spectral measure p on X into A; by Lemma 1, we can assume without loss of generality that p is one-toonto A such that, if a = one. Thus p is an algebraic isomorphism of p(f), r(a) = If II (Theorem 3). It follows that a r(a) is a norm on A with into respect to which A is a Banach algebra; since p is continuous on A, this norm topology is finer than the (induced) topology of A. Hence if a —* r(a) is continuous on A, the two topologies are identical and p is a topological isomorphisrn.
then any isomorphism (b) (a). If A is equivalent with some establishing this equivalence is a spectral measure; the remainder follows from Theorem 3. (a)
(c).
complete.
is This follows from (a) (b) and the fact that every The situation that a r(a) is continuous presents itself, in par-
ticular., when A is a normed algebra since r(a)
Ia
in this case.
COROLLARY... Every spectral subalgebra of a normed algebra is complete.
It is an open problem to characterize the closure in A of a spectral subalgebra A on which the spectral radius is not a continuous function of a. It follows from the definition of a spectral algebra and Theorem 1 that each spectral algebra can be ordered in such a way that it becomes the (real or complex, accordingly as A is real or complex) linear hull of its unit interval J = (a: 0 a e}. To within a completeness condition, this property characterizes spectral algebras (and hence algebras of all continuous scalar functions on a compact space). THEOREM 6. Let A be a-commutative, ordered locally convex algebra which is the linear hull of its unit interval J; I is assu,ned to be semi-complete. Then A is a spectral algebra. PROOF. It follows from the hypothesis that A = A1 + iA1 where A1 = Ur n[—e, e] (the interval [—e, e] is equal to 2J — e). For each a e A1 there exists, by Theorem 4 and its second corollary, a spectral measure on c(a) c R which is positive for the given order of A. Denote by X the compact ae A1}. Since the family (Wa} is clearly abelian, Theorem 2 imspace fl plies that the product measure p ® exists on X into A (obviously the required semi-completeness of J can replace the semi-completeness of A in the present circumstances). Since a = for every a e A1, it follows that a = p(f,,) when denotes the projection of X onto X4 which is a continuous real-valued function on X. Thus if a + ib A (a, b A1), it follows that a + ib = + ifb) and the proof is complete. By combining Theorem 6 and the type of argument used in the proof of Theorem 5, one obtains what appears to be a new characterization of the
H. H. SCHAEFER
468
for compact X. Let A be a commutative (real or complex) Banach algebra with THEOREM 7. and only if it is capable of an ordering unit. A is equivalent to some under which it is the linear hull of its unit interval J. Such an ordering is necessarily a lattice ordering'0 of A. PRooF. The necessity of the condition is obvious. If the condition is satisalgebras
fied, Theorem 6 implies that A is a spectral algebra, i.e., there exists a spectral measure p on a compact space X onto A. By Lemma 1, p can be assumed to be an algebraic isomorphism; the continuity of a —o r(a) implies that p is a homeomorphism. From Corollary 1 of Theorem 4 and Theorem
3 it follows now that p is an order isomorphism (for the natural order of and the given order of A) and the proof is complete. REMARK. It is an unsolved question whether in Theorem 7, the word "commutative" can be omitted.* More generally, is a real algebra consisting entirely of real spectral elements necessarily commutative? With the aid of Theorem 6, it is not difficult to verify that every real algebra of Hermitian
operators on a Hilbei-t space H which is closed for the uniform operator similarly, every closed topology, is a real spectral subalgebra of *.algebra of normal operators on H is a (complex) spectral subalgebra of .9'(H). What examples are there other than those classical ones? We wish to quote, without proof, the following result whose details have been carried out in [12, §4]. Let E denote a semi-complete, locally convex vector lattice on which every the algebra of weakly positive linear form is continuous; denote by Continuous endomorphisms of E under the topology of simple
and by / the unit interval of .5t'(E) for the induced order. (These assumpThen tions imply that J is semi-complete, and an abelian subset of the linear hull of J is a spectral subalgebra of .2'(E). If, moreover, T -.-' 0 implies r(T)—.O, then this algebra is a complete normable algebra under the induced topology, and hence closed in Sf(E). For instance, it follows from the preceding result that the closed subalgebra of that contains all continuous endomorphisms of L,(O, 1) which have a diagonal matrix representation with respect to a fixed unconditional basis of (p > 1) is spectral. Moreover, with the aid of [7, (11.3)] it can (in
be shown that the algebras of Hermitian and normal operators mentioned above are contained in spectral algebras of the class considered in the previous paragraph. 4. Extension of spectral measures. Spectral elements of a locally convex
algebra A are defined as elements imbeddable in certain commutative subalgebras which are, essentially, algebras of continuous scalar functions on a compact space X; the spectral simplicity of these algebras, largely due to the absence of a radical, is then reflected in the spectral behavior of their 10
In
the complex case, a lattice ordering of K — K (K the positive cone in A). in proof. B. J. Walsh has answered this question affirmatively.
* Added
469
CONVEX CONES AND SPECTRAL THEORY
elements. For example, if a is a spectral element, then every isolated point
of 0(a) is a pole of the resolvent of order one, and c(a) = {O} implies a = 0. When A is an algebra of operators, spectral elements have spectral adjoints. However, to obtain the full range of re%ults familiar from the spectral de-
composition of Hermitian and normal operators in Hubert space, a somewhat more elaborate description (namely, in terms of projections) of a spectral operator is needed. We discuss in this section briefly the extension of spectral measures that makes such a description possible. Let S denote a normal (bounded) operator on Hilbert space. The spectral behavior of S is completely characterized by its spectral representation S=
AdP(A);
here .9: 8
P(8) is a countably additive and multiplicativeU mapping, with P(C) = I, of the Borel sets of the extended complex plane C into the set of orthogonal projections on H (under its strong topology). .9 is called the reso-
It is an imto take its values among the orthogonal projections of some Hubert space H; all that is needed to reproduce the classical theory of spectral representation is to construct, for a given spectral element of a locally convex algebra A, a resolution of the unit of A with the above properties but taking its values among the idempotents of A. It will be shown that, under comparatively mild completeness assumptions, every spectral measure with values in A gives rise to a resolution of unity. Conversely, if .9 is a resolution of unity defined on the Baire sets of a compact space X and with values in A, then lution of the identity (or spectral measure) associated with S.
portant realization to see that it is entirely unessential for .9
f—i.
(1€
is a spectral measure on X into A such that p(l) = e. Let A be a locally convex algebra, p a spectral measure on a compact space X into A. For greater convenience of presentation, we assume that A is a real algebra and a real spectral measure. Recall that the Baire (Borel) sets in K are the members of the smallest class of subsets of X that contains all closed sets of type C8 (all closed sets) in X and is invariant under the formation of symmetric differences and countable unions. A (finite) real-valued function on X is a Baire (BoreD function if its inverse f' maps the a-algebra of Borel sets in It into the a-algebra of Baire (Borel) subsets of X. Denote by
the algebra of bounded Baire (Borel) functions on X, and
by A the weak completion of A. A is a locally convex space with respect to tha weak topology u(A, A').
Denote by K the positive cone of A for any order of A with respect to which p is positive (Theorem 1, corollary); since K is weakly normal, A' = K' K' where K' is the dual cone of K. If co€ A', then p,(f) (p(f), is a measure on X [1]; denote by its extension in the sense of [11 to a'(X). is multiplicative if P(ô fl e) = P(ö)P(e) for arbitrary 8,
H. H. SCHAEFER
470
with the locally convex topology
We provide
generated by the
semi-norms (ço€K')
(this topology is, in general, not Hausdorif). It is easy to verify that 'e(X) into A for the and that p is continuous on for is dense in and a(A, A'). Thus p has a unique continuous extension to topologies with values in A that is obviously linear. If it is known that maps into A, then it follows from the separate continuity of multiand A (under the topologies presently considered) that plication in into A since p is a homomorphism of is a homomorphism of
Assume now that this is the case and, in addition, that 4a(1) = e. Then the set function o—*p(8) =
a resolution of the identity defined on the Baire (Borel) sets of X into A, is a homomorphism of the and said to be generated by p. It is clear that is
Boolean c-algebra of Baire (Borel) sets in X onto a Boolean cialgebra of idempotents in A, countably additive for c(A, A'). The following theorem into A, and actually maps gives conditions under which countably additive for topologies other than o(A, A'). THEOREM. 8. Let p be a spectral measure on X into A such that p(l) = denote by K the positive cone of A for the finest order with respect to which p is positive, and by J the corresponding unit interval in A. If J is weakly into A, and the maps semi-complete (weakly complete), then under which
resolution
is
the identity generated by p is countably additive for every consistent
topology on A under which K is normal. PROOF. We assume first that J is weakly complete. If we denote by I the fl I is dense in I for the 1) of unit interval {f: 0 then fl I) c J; this implies /1(1) c 3 and, topology Z considered above, and is the linear hull of I, that the range of /1 is contained in A. since
If I is only semi-complete for the weak topology, we proceed as follows. Let If,,) c I be a sequence such that limf,(t) = f(t) for all t X. Since {f,,) {,ü(f,,)} is a weak Cauchy sequence in A and is a Cauchy sequence for c J implies /1(f) EJ. Hence In is closed under the formation of simple limits of sequences. Since /r'(J) contains all continuous functions in I, it follows that it contains the characteristic functions of arbitrary Baire sets in X and, consequently, that is mapped into A by ,a. Finally, the assertion concerning the countable additivity of follows from its countable additivity for the weak topology c(A, A') and from Theorem C of the preliminary section. This completes the proof. Furthermore, Theorem C implies that when J is weakly complete, the limits ji(z,.) = /1(Za) = limG /1(Z9)
exist for every Baire (Borel) set ö c X, with respect to every consistent
CONVEX CONES AND SPECTRAL THEORY
471
stand for the directed topology on A for which K is normal. (Here sets of open (closed) subsets of X containing (contained in) 8.) We remark in conclusion that resolutions of the identity make it possible to integrate essentially unbounded functions with respect to a spectral measure, and thus to discuss spectral operators on locally convex spaces with unbounded spectrum. The details have been carried out in E12]. REFERENCES
1. N. Bourbaki, Integration, Actualités Sd. md. No. 1175, 1244, 1281, Hermann, Paris, 1952, 1956, 1959; Chapters 1•IV, V, VI. 2. F. F. Bonsall, Endomorphisms of a partially ordered vector space without order
unit, J. London Math. Soc. 30 (1955), 3. , Linear operators in corn piete positive cones, Proc. London. Math. Soc. (3) 8 (1958), 53-75.
4. S. Karlin, Positive operators, J. Math. Mech. 8 (1959), 907-937. 5. M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Mat. Nauk 3 (1948), no. 1 (23), 3-95 = Amer. Math. Soc. Transi. No. 26, 1950.
6. H. Schaefer, Haibgeordnete lokolkonvexe Vektorraume, Math. Ann. 135 (1958), 115-141.
7.
,
Haibgeordnete
lokalkonvexe Vektorräume. II, Math. Ann. 138 (1959),
,
Haibgeordnete
iokalkonvexe Vektorrdume. 111, Math. Ann. 141
259-286.
8.
(1960),
113-142.
9. 10.
10
On non-linear positive operators, Pacific J. Math. 9 (1959), 847-860. , Some spectral properties of positive linear operators, Pacific I. Math. (1960). 1009-1019. ,
11. , On the singularities of an analytic function with values in a Banach space, Arch. Math. 11 (1960), 40-43. 12. , Spectral measures in locally convex algebras, Acts. Math. 107 (1962), 125-173. UNIVERSITY OF MICHIGAN
THE DUAL CONE AND HELLY TYPE THEOREMS BY
F. A. VALENTINE 1.
Introduction. A significant contribution to the theory of convex sets
was made by Minkowski when he introduced the support functional [24; 25]. The dual cone, in recent years, has enabled geometers to appreciate in clear spatial form the nature of the support functional of a convex set. The technique used here is similar to that used by Dieudonnê [3], Klee [19) and others in their proofs of the Hahn-Banach theorem. The same technique was used their support functionals by Fenchel in the characterization of convex sets [5]. Also see Hille and Phillips [14]. A more extensive use of this method was made by Sandgren [33] in his study of certain Helly type theorems. It is perhaps significant to mention that we discovered Sandgren's results before reading his paper. Our paper includes refinements of known results, together with a number of further applications. For instance, parts of the proof of Theorem 12 are new, and Theorem 15 is new. Each theorem in 2 and 3 is preceded by a historical acknowledgement. Before developing the main theory, a few well-known elementary facts will be described in essential detail. The reader who is familiar with the papers mentioned may very well by-pass this first section.
Although we introduce the theory for sets in Euclidean rspace E, many of the results hold equally well in a complete inner product space. In order to describe matters easily, we use the following notations: Notation. In the following, x• u denotes the usual inner product of x e E, u E, where E is the space. The closure, interior, boundary and convex hull of a set S c E are denoted by ci S, mt S, bd S and cony S. respectively. The closed line segment joining x e E and y E E is indicated by xy, whereas L(x, y) denotes the line determined by x and y, if x * y. The interior of a set S c E relative to the minimal flat containing it is denoted by intv S. Set union, intersection and difference are U, (1 and respectively. We let 0 and 0 stand for the origin of E and the empty set, respectively. Finally, x + y denotes vector addition where xe E, ye E, and Ax denotes scalar multiplication, x e E, 2€ being the real field. We assume the set M 0 except for Theorem 5. DEFINITION 1. If M is a convex set in E then the real-valued functional h is the set of all those pairs Eu, h(u)], u E, such that h(u) = supx-u < co.
functional h is called the support functional of M. The support functional satisfies the following well-known properties [1).
The
473
F. A. VALENTINE
474
THEOREM l. The domain of definition D of a supPort functional h given by (1) is a convex cone having the origin 0 as an apex.
The function h is a continuous, subadditive and positively honwgeneous'functional for all u E D, so that h(Au)
(2)
Ah(u)
,
D, u€ D, v€D.
A
h(u + v) < h(u) + h(v),
0, u E
Theorem 1 is an immediate consequence of Definition 1. For instance, if
ueD, veD,
we have
+ v)
XEM
x€.V
sup Ax.u = xEM
+ supx•v < 00, ZEM
reM
so that u ÷ v e D, Au e D, so that (2) holds.
It should be observed that if a point x0e M exists such that h(u) = x0•u, then = h(u) is the equation of a plane of support to M at x0, since (1) implies
x€M. The continuity of h follows from the convexity of It. THEOREM 2.
If M is a bounded closed convex set in E, then for each u e E
there exists a point x0 M such that h(u) = x0 . u.
This is a consequence of the compactness of M. In the infinite dimensional case, the so-called weak compactness implies the corresponding result. REMARK 1. If M has the support functional It defined on D, then a translate M + x1 of M has a support functional It1 defined on D such that
ueD. THEOREM 3. If M is a nonempty closed convex set in E with supPort functwnal It defined on a nonempty domain D then M satisfies the condition
h(u),uGDJ.
To prove Theorem 3, observe that the set H,. [x: x'u h(u), u fixed in DJ is a closed half-space. Since M c H,. for each u D, we have
McflH,,. ,.E 0 there exists a point z H,.) M * 0. Without loss of generality, assume 0 M. Since M is closed, there exists a point w = Az (0 < A < 1) such that w M. Since M is a closed convex set, and since the space E is locally convex [19], there exists a hyperplane H such that H, B ri M = 0. Hence, H strictly separates Mand z, since w = Az, 0 < A < I, and since 0 €M. Suppose
Thus, there exists a vector
whose corresponding segment from 0 is perpendi-
cular to H and intersects H. Let H,' denote the closed half-space bounded
THE DUAL CONE AND HELLY TYPE THEOREMS
475
< so that by H which contains M. Since M c we have SUPXEM e D. However, since z H:, we have z IL, a contradiction.
Hence, (3) holds. DEFINITION 2. A functional h satisfying condition (2) for all u on a convex
cone D c E is called a support functional. We are now in a position to develop the main theme of this paper. 2. The dual cone. As mentioned in the introduction, the principle of the dual cone has been used by Fenchel [5], Sandgren [331, and it plays an increasingly important role in modern functional analysis, Köthe [22]. It was essentially used also by Rademacher and Schoenberg [29] in their simple proofs of two theorems of Kirchberger [18]. It is the opinion of this writer that the principle of duality used here has only begun to be exploited. No other concept has influenced us as much in recent years, and my colleagues have patiently endured my sustained enthusiasm, for which I am indeed grateful. is the real field, let If E is a Euclidean space and E) denote the usual product space with the corresponding product topology. The reader may wish to carry through the corresponding theory when E is a complete inner product space, which can be done with no difficulty. If E = E2, then E) may be chosen to be E3. This model may be useful to the reader unfamiliar with the theory. The axis (z, 0) will be called the vertical z axis
of
E). DEFINITION 3.
Let M be a convex set in a Euclidean space P2, whose support
functional h is defined on D. Points (z, u) e
Then the dual cone C of M is the set of all
E) such that
C=[(z,u):z h(u),ueD]. Although we will be primarily interested in the case when D is a closed
cone, there are instances when this is undesirable. However, if M is bounded and closed, then C has the following preferred form. THEOREM 4.
If M is a closed bounded convex set in E, then the dual cone
E) defined for all u E, and having the (5) is a closed convex cone in E) as its vertex (0 is the origin of P2). The vertical ray origin (0, 0) e [(z, 0), z > 0] C mt C. PROOF. The boundedness of M implies the inequality in (1) holds for all P2, so that h is defined for all u. Since h is a convex functional, the set C is convex. The positive homogeneity of h implies that C is a cone with (0, 0) as vertex. Clearly C is closed in the product topology. Also since u
(I, ø)eintC, clearly (z, ø)eintC, if z>0.
As shown by Fenchel [5], convex sets can be characterized so simply by their support functionals. There is no need to resort to the Gateaux differential [1] as was done in the past. The following proof is due to Fenchel 15]. THEOREM 5.
Let h be a
functional defined
a closed convex cone
F. A. VALENTINE
476
D * 0 having 0 as an apex (see Definition 2). Then the set of points M= [x:x.u h(u), all u€D] convex set.
is a PROOF.
It is not the convexity of M
which
ft=rl,u€D
is difficult to prove, for if
+ Bh(u) = h(u), so that ax + $y e M. Hence the only real difficulty is to prove M is not empty. Consider the dual cone C given by (5). Since C is a closed convex cone containing the point (1, 0), and since (—1, 0) C, a fundamental separation E) exists which contains e theorem [19] implies a closed hyperplane E) being locally convex. The translate (—1, 0) and misses C, the space Hof which contains the vertex (0, 0) of C supports C such that (1, 0) H. E) the hyperplane H has the form Since (1, 0) H, in ah(u)
H=[(z,u):z=xo.u,ueE). Since
H supports
C, we have
The proof where E is a complete inner x0 M, and thus M * 0. product space is essentially the same, with a few modifications similar to the corresponding proof for the Hahn-Banach theorem [191. REMARK 2. Let E" be a closed flat in E of deficiency n. The dual cone of is a cone whose domain of definition is a subspace of dimension n. (The deficiency of E" is the difference of the dimension of E and of E*.) This well-known and obvious result is a consequence of the fact that the condition in (1) holds only for those directions which are perpendicular to and these directions yield a flat of dimension n. In order to derive simple proof s for Helly type theorems, the following significant modifications of the theorem of Carathéodory [21 are given. THEOREM 6. Let S be a set in a linear space L. Choose an arbitrary point v e S. Then x e cony S sf and only if x is contained in a finite dimensional simplex having its vertices in S, and having v as one its vertices. PROOF. The proof follows conventional lines. See [1). THEOREM 7.
Let
(i = 1,
n)
be n convex sets in a linear
space L and
let
Then x E cony
v,xj,xt, ...,x. in Sufficiency.
if and only lix €4, where 4 is a simplex with vertices
and where s
n.
By Theorem 6, there exists a finite dimensional simplex 4 and containing x. If N> n, by a in U= x, , x2, •,
with vertices v,
relabeling x1 E
C1,
of subscripts there exist two vertices, say x1 and such that C1. Since C1 is convex, it is simple to verify that we can replace x1x2 such that XE COflV (v U z U x3 u •.. u XN). By
x1 and x2 by a point z
THE DUAL CONE AND HELLY TYPE THEOREMS
477
induction, the stated conclusion follows. Generalizations of theorems of Carathéodory type have been recently presented by my colleague Professor Motzkin in a colloquium address before the American Mathematical Society [271.
The two following theorems are similar to the theorems of Sandgren [33] and are obtained from Theorems 6 and 7. THEOREM 8. Let {M,, i e A} be a family of convex sets in E. If they all have a point in common then all the dual cones C, corresponding to M., i A, have a common nonvertical hyperplane of support in
ueE].
E) of the form [(z, u): z —
PRooF. Choose x0 M, so that h,(u) for all u eD,, 1€ A. This implies that the hyperplane E(z, u): z = ue El in E) bounds all the cones C given by (5) with h replaced by h,. It is clearly nonvertical in E). When the sets M4 are bounded and closed, it is of particular interest to obtain Sandgren's theorem [33] by a significant application of Theorem 6. THEOREM 9. Let {M1, i A) be a family of bounded closed convex sets in E. They all have a Point in common if and only jf all the open cones, mt C,, i A, have a common nonvertical hyperplane of support [(z, ii): z = x0 U, u E E] in (a', E), where C, is the dual cone of M1. This is equivalent to the condition
(6)
conv(UintC1)
PROOF. A. First, suppose fl, M1 * 0, where, for simplicity of notation, the index i ranges over A. This implies a common nonvertical hyperplane of
support
H= [(z,u),z = xo.u,ueE] to all the cones C , i A, exists. Clearly H also supports each mt C,, i e A, at (0, 0). Conversely, if H supports each mt C, i A, it also supports each C,, i A, since C1 = clint C,, so that •u h(u) for all u E, i A, and xoefl1M1. B.
To prove the necessity of (6), suppose (6) fails. Since (1, Ø)€ mt C1, ie A,
Theorem 6 of Carathéodory implies there exists an s-dimensional simplex 4
having its vertices in U mt C, having (1, 0) as one of its vertices, and such that (0, 0)64. Since each vertex of 4 is in mt C,, and since (1, 0) E mt C1, i e A, each hyperplane in E) through (0, 0) must intersect at least one of the open cones mt C,, 1€ A. However, this violates the result of the preceding paragraph A. Conversely, suppose (6) holds. This implies (0, 0) bd cony (Li1 mt C1). A nonvertical hyperplane H of support to cony (U1 mt C1) through (0, 0) exists. Hence, paragraph A above implies fl1 M, * 0. Again we use the Carathéodory type Theorem 7 to yield very simply the
F. A. VALENTINE
478
following theorem of Lanner [23J. in a recent letter Professor Folke Lanner informed me that Professor F. Riesz (311 in a letter to his brother Professor M. Riesz described a proof essentially the same as that which follows. ., n) be n compact convex sets in a Euclidean TSEOREM 10. Let M, (i = sfrace E, and let h, denote the support function of Mi (i = 1, ., (i = .. n) have Then a necessary and sufficient condition that the sets a common Point is that
(7) holds for each set of points u (i =
•, n) satisfying
1,
Sufficiency. We prove the sufficiency by proving its contrapositive. Suppose M, = 0. Then Theorem 9 implies
E).
=
(ci
E) is contained in a simwhere v = (0, 0), where s n, and C (k = 1,. .•, s). The proof of Theorem 7 implies that we may, without Xk loss of generality, relabel subscripts and choose s so that Xk Ck (k = 1, . , s). Since E), let Xk (z&, E)(k = 1, s). Since
Theorem 7 implies that the point (—1, 0) e
plex 4 having vertices
v, X1,
•, x1
•
(—1, O)€conv(v, x1, there exists constants A,
> 0, such that
0,
(—1,0) =
v1)
where s + 1. v,) = (0, 0) and A, = 0, if (0, 0) as a common vertex, we have U, Ay, dition (9) implies —1
=
0
A,e,
Since C, are cones, having
A,h,(v,) =
Since
h,(v,), con-
E h,(u,)
ru,.
Since (10) violates (7) and (8), conditions (7) and (8) imply proves the sufficiency. Necessity. This is trivial, for x0 M, * 0 implies
M, * 0. This
x0.u,_h,(u,) for all u,€E, whence (8) implies (7). 3.
ilelly type theorem8. We now establish 1-lelly's theorem [121, this time
479
THE DUAL CONE AND HELLY TYPE THEOREMS
appealing to Carathéodory's Theorem 6. (See also Sandgren [33].) THEOREM 11.
Let
be
a family of compact convex sets in Euclidean n-space
having at least n + 1 members. A necessary and sufficient condition that all the members of 5 have a common point is that every n + 1 members of 9- have a common point. is finite, then the word "compact" can be If the number of members of omitted to obtain the same conclusion. PROOF. Let 9- = {M,,,, a A}. To prove the sufficiency of the condition = 0. Theorem 9 when M0 , a A, are compact convex sets, suppose flIEA where C. is the dual cone of M.. Since implies cony (U.€4 mt C0) =
(1, 0) mt c. , a A, Carathéodory's Theorem 6 on simplices implies that X.+I , s n, such (0, 0) €4, where 4 is a simp!ex with vertices v, x1 , x2, that E U.€A mt C.. Without loss of generality, relabel subscripts so that e mt C, (i = 1, . , s + 1). This, together with the fact v (1, ø)e mt C,,, a€A, implies that each hyperplane through (0, must intersect (i = 1, - -, s + 1) and hence at least one of the at least one of the edges -, s + 1). Since s + 1 n + 1, this contradicts the (i = open cones mt fact that the s + 1 cones C (i = •, s + 1) have a common plane of support M, = 0, a contradiction. at (0, 0) in Es), which in turn implies that Since the necessity is obvious, this completes this part of the proof. When the set .F has a finite membership, it is a simple matter [29] to prove the existence of compact convex polyhedera P0, a A, such that P. c M., and such that every n + 1 members of {P., a A} have a common point, thus reducing the situation to the compact case. We now state and prove a theorem which exhibits in fine form the power of duality. The implication (ii) (iii) is due to Horn [16] (see Sandgren [33] for a similar proof using duality). For n = 2, it was first proved by Horn and Valentine [15]. Our proof differs from Sandgren's in that we use an elementary theorem proved by Hanner and Radström [11]. The implications (iii) -- (iv) and (iv) —÷ (ii) are due to Klee [21]. The dual reasoning used here implies these results with remarkable simplicity. The implication (1) —' (ii) (Valentine) has been added as it is significant, and the proof of Klee's result v
- -
- .
-
-
(ii) is simplified. Shapley 117]. (iv) —p
For another proof of
(ii)
(iii) see Karlin and
THEOREM 12. Let ..9 = {M,,, a A} be a family of bounded closed convex sets in E. The following four statements are equivalent, where n is a positive integer, and where contains at least n members.
(I) n— 1 (See
For each set of n members of 9 and for each flat
of deficiency
there exists a translate of E"' which intersects these n members of 9-.
Remark 2.)
(ii)
Every n members of 5 have a point in comnion.
Every flat of deficiency n is contained in a flat of deficiency n — 1 which intersects every member of (iv) Every fiat of deficiency n — 1 has a translate which intersects every (iii)
F. A. VALENTINE
480
member of PRooF THAT (1)
(ii).
Suppose this is false.
This and Theorem 9 implies intC,) = ,n), such that By Theorem 7, there exists an
there exists n members of is the dual cone of E), where having mt s-dimensional simplex 4 with $ n having its vertices in E). (1, 0) v as one of its vertices, and containing the origin (0, 0) of E) which contains denote an n-dimensional subspace of I et denote the subspace of E of deficiency n — 1 whose support 4. Now let is a Hence, the set ((z, u): z = h(u), u function h is defined on say M1 (i =
-
E has a linear subspace of the space (a', E,,_1). A translate support function h1 which also has E._1 as its domain of definition. Since 1€ A, (0, 0) e 4, since each vertex of 4 is an interior point of some cone and since v (1, Ø)eintC0 for all aEA, the (n —1)-dimensional hyperplane must intersect at least one of the open in [(z, u): z = hj(u), u e n, and u): z = cones mt (i = 1, •--, n). Hence some cone, say c,, I j E). Dually, h1(u), u e E,,_11 cannot have a common plane of support in of cannot intersect M,. Since this this implies that the translate holds for each translate of the hypothesis (i) is violated, and hence (i) (ii) has been established. PRooF THAT (ii) (iii). In order to prove this we first state the following theorem of Hanner and Radstrom [11]. Let S be a compact set in an ndimensional Euclidean space Suppose p is a Point of & for which no s points of S exist with s n whose convex hull contains p. Then there exists a hyperplane H through p which does not intersect S. To prove (ii) —' (iii) let E' be a flat of deficiency n in E and having a supbe the nport function h defined on a subspace & of dimension n. Let dimensional hyperplane in E.) defined as follows: (11)
H,,
[(z, u): z = h(u), u E E,j -
To prove that (ii) (iii) it suffices to establish the existence of an (n — dimensional subspace H,,_1 of H,, such that
1)-
(12)
for each M,, 5, where C. is the dual cone of M.. We prove (12) for the case when 9 has finite since the infinite case follows readily. To accomplish this let (13)
C. for which H,, n mt = 0, so let A, denote the subset of A such that H,, n mt C. 0, a e A1. Let the interior of relative to H,, be denoted by Tnt so that we have
aEA,. Firstly, if (14)
/
* H,,
481
THE DUAL CONE AND HELLY TYPE THEOREMS
then any (n — 1)-dimensional hyperplane of support in H,, to B through (0, 0) is the desired H,,_1, since it satisfies (12). Secondly, if
conv(U IntK.) = H,,, .E mt K. * 0 for a A2, there exist closed convex cones a E A1, all having 0 as a common vertex such that mt since
c 0 U mt K.,
0, a G A1, and where K. is as defined in (13). By mapping the such that K. as i —i sets J, on the unit sphere of H,, with center at (0, 0), the theorem of Hanner and Radström [11] preceding condition (11) implies the following: (a) either the origin (0, 0) in H,, is contained in a simplex 4 having at most n vertices belonging to UII€A1 0) and hence belonging to U.€4, mt K or for each 1(1= 1, 2, of H,, such that (b) there exists an (n — 1)-dimensional hyperplane A1. 0) for all H,_1n J,,'= (0,
In case (a), since the vertices of 4 belong to UOIEA1 mt C., and since E) through (0,0) could support (1, O)EintC.,aEA, no hyperplane of those n or fewer cones of the set (C.. a A} which contain the vertices of 4. Dually, this contradicts (ii).
In case (b), since the sets K, are locally compact, the sequence of hyperhas a convergent subsequence which converges to an (n — 1)planes dimensional hyperplane H,,_1 c H,, satisfying (12). Let h1 be the linear function of H,,-2 defined on a subspace E,,_1 of E. Then
E'1
[x: x-u
h1(u), u
E,,_1]
1 which contains E', and which intersects every member of 9—. This completes the proof of (ii) (iii). PROOF THAT (iii) (iv). Let E'1 be a variety of deficiency n — 1 > 0 with support function h defined on E,,_1. Let E,, be any n-dimensional subspace
is a flat of deficiency n —
of E containing E,,_1, and choose x e E,,
sional subspace of space E,,-1, where k = dimensional subspace
E,,)
2,
E,,_1.
Let E,,k denote the n-dimen-
which contains the point (1, 0) + x/k and the
3, . -.. Hypothesis (iii) implies E,,k contains an (n — 1)E,,) which misses each open cone mt Ce,, of
E,,) is finite dimensional, a convergent subsea e A. Since the space quence of exists which converges to a space called Also since ri E,,) is locally compact, the space E,,1_1 also misses each c 00, we have open cone mt C,,, a A. Since Ek,, E,,_1) as k E,,_1).
The linear functional h2 of
is defined on E,,_1, so that
[(z, u): z = h1(u), u
E,,_2]
-
flat E1111 in E having h1 as its support function is a translate of E'1 which intersects every member of 9-. Since (iv) — (i) is trivial, we have completed the proof of the equivalence of the four statements. The
The following theorem of De Santis [34] has a particularly elementary proof
F. A. VALENTINE
482
when examined dually. of convex THEOREM 13. If every k + 1 or fewer members of a finite family members of sets in .E contain a common flat of deficiency k, then all the contain a common flat of deficiency Fe. •, s) denote the member of .5, and let D1 denote PROOF. Let M. (i = 1, of M1 (see Theorem 1). the domain of definition of the support function The dual of the hypothesis in Theorem 13 implies that every Fe + 1 of the cones D1 (i = 1, .. ., s) are all contained in a linear subspace of dimension k. This fact implies that
(JD. is contained contains at most k linearly independent vectors, so that should (In this proof, the domains in a linear subspace of dimension k. The dual of the not be confused with the dual cones C, defined in (5).) contains a 1, . •, s) foregoing implies that every Fe + 1 member of M1 (i = common flat of deficiency Fe which is a translate of a fixed space of deficiency in Ek. Let be the set of Fe, denoted by Ek. Choose a basis ej , e2, ••, is convex. which are parallel to e,. Clearly each all lines belonging to Let Ek be the orthogonal complement of Ek. Applying Helly's theorem to for each fixed .j = 1, , N, we immediately obtain the the sets K1, fl
existence of N independent lines in fl
M1,
whence fl..i M contains a flat
of deficiency k.
We now go on to investigate the existence of common transversals and common intersecting planes. For a recent thorough summary of results on these matters see GrUnbaum [8]. We first present a new proof for a known
result for sets in E, and we then present a new result for E1 which no doubt can be extended to in exactly the same way. The following theorem was discovered by Klee [20] and Grunbaum [7] independently.
Suppose .9 is a family of compact convex sets in the Plane Also suppose there exists a line L c such that for each Pair of members in there exists a translate of L which strictly separates these two sets. If every three members of are intersected by a common line in E,, then all the members of .? are intersected by a common line. PROOF. In order to prove this we use the following concept. DEFINFrI0N 4. If C, is the dual cone for the compact convex set M1, then let C denote the symmetric image of with respect to the apex (0; 0) of C and relabel C1 so that C' U C:, is the two napped cone determined by It should be observed that C,' n C,! is either a point, a line or a plane in THEOREM 14.
E2.
E2).
If M4 and M1 are two distinct members of
then, since M, ii M, = 0,
Theorem 9 implies (15)
cony (mt C1 U mt C,) =
Es).
THE DUAL CONE AND HELLY TYPE THEOREMS
483
and M1, Let L which by hypothesis strictly separates is defined on only a and let the support function of L" be Clearly one-dimensional subspace of E2, denoted by E1,. Since L" is a translate of L for each pair i and j, we have (16)
E1
and E1 is independent of i and j. The set
E
L,
(17)
u E,)
[(z, u): z =
is a line in the vertical plane E1) c (a', E,) through (0, 0). that strictly separates and M1 implies we have (18)
— (0,
0) c (mt
The fact
U (mt
Hence, Definition 4 implies (19)
0) c mt (C n CJ). K. denote the set of all lines in E2) through (0, 0)
L1, — (0,
DEFINITION 5. Let
which miss mt C. To prove Theorem 14, we merely need to prove the dual conclusion
fl K,*(0,O).
(20)
.9
To do this choose a point p E2 — + p. namely,
so that the product space of
and
is a plane parallel to (a', E1) through p. Then the intersection of
with
E1
this plane, namely
+/'), is
a closed connected set lying between two open disjoint convex sets mt C1(p) where
and
Cr(p) Cr fl E1 + P) Let (a = 1,2) be a closed half-line of the vertical line which is in the interior of Cr(p) n C7(p). Similarly let
(a
p)
=
= 1,2). 0) + p
and R.' be two half-lines of the line L1, + p which are in the interiors of C(p) n C(p) and C(p) n respectively, where L4, + p is the translate of L,, passing through p. Let the endpoints of (a, = 1,2, a be denoted by x?f, respectively. Since
cony (R?, u
c Ct(p), c Cf(p),
(a,
we have 1(4 n K1 n (R, E1 + p) c
u
so that the left member of (21) is a compact set.
U
U
= 1,2, a *
484
F. A. VALENTINE
The left member of (21) is also a cell (homotopic to zero in itself) [8]. To x) in the plane prove this, first observe that each vertical line + P)
intersects K1 n K5 in a segment or point. In terms of the usual positive E1 + p), this implies that the boundary orientation of vertical lines of E1 + /') relative to E1 + p) can be decomposed into an of K n K, n upper and lower part, each of which is an arc. Since mt C,'(p) n mt C?(p) = 0 (i = 1, 2), the upper part consists of two convex arcs or points which belong to bd and bd CJ(p), respectively, whereas the lower part belongs to two convex arcs or points belonging to lxi C(p) and CJ(p), respectively. The first two arcs are concave upwardly, and the last two concave downwardly. The upper and lower arcs may intersect, but they do not cross. In any case, it is quite clear that these facts imply K, n K5 n E! + p) is a cell. This
fact may be proved in other ways, and a combinatorial procedure is given in Theorem 15 for higher dimensions. To prove (20) holds, it suffices to prove it for each finite subcollection .9'T of .9. Let S be infinite right circular cylinder having the vertical line p) as its axis of symmetry, and such that it contains all of the compact sets K1 fl K5 fl E1 + p) for M1 e M, e It is obvious that + p) are cells for M. e flL since mt C,'(p) n mt C(P) = 0. K. n S n Also since each line in K1 n K5 through (0, 0) must intersect E1 + p), the hypothesis of the theorem implies
÷p)*0 for each triple
M1, Mk in Since the sets in (21) and K, n S n E+ p) are cells, condition (22) implies, by Helly's generalized [8; 131 on intersecting cells, that (20) holds for .9-i. Compactness then implies
(20) holds for .9-. The dual of (20) implies a line exists in E2 which intersects all the members of This completes the proof. We are now in a position to generalize Theorem 14 to E3, and the method used will extend to E,,. THEOREM 15. Let 5 be a family of compact convex sets in E1. Also suppose there exist three distinct planes P (i = 1, 2, 3) in E3 containing a common line such that for each triple of members M1, M1, M3 in 5 each pair of the triple is strictly separated from the remaining member of the triple by a trans late of either P1 or P2 or P8. and this correspondence is cyclic. If every four members of .9 have a common intersecting plane, then all the members of .9- have a common intersecting plane.
For three sets M1 (i = 1,2, 3) in .9-, without loss of generality, we may rearrange the subscripts on (i 1, 2, 3) so that there exist translates of I'd, denoted by + P1. such that M1 and cony (M2 U M2) are strictly separated by P1 + p1 , M2 and cony (M8 U M1) by P2 + P2, A'!3 and cony U M2) by P8 + p - The permutation of indices is cyclic, and we will indicate the
range (1,2,3) of the indices only when necessary. Since the planes {P3 (i = 1,2,3)) have a common line, the support function h, of P is defined on a line A, and A c E3 (i = 1,2,3), where E8 is a plane
485
THE DUAL CONE AND HELLY TYPE THEOREMS
of
E3 independent of i. Also E2
+ p, in the extended space
the dual cone of
As a consequence,
cony (D1 u D2 u A). E5)
is a vertical half-
plane bounded by the line u D1, a = [(z, u): z = is the support functional of P1 + where In order to prove Theorem 15, it is sufficient to show that (20) holds when EL has been replaced by E3. We will show that (20) holds for each finite (i = 1,2,3). Since, by of the family 9. Choose M, c subfamily agreement, P1 + strictly separates M1 from cony (M1 u Mk) (i, j, k = 1, 2, 3, i L1
,
j * k), we may translate the three planes P, + p1 so that the three lines and L3 in (23) are not coplanar, and still have the separation property
relative to M1, M, and M8. Moreover, there exist translates P1 + P1 (i = 1, 2, 3) of P, which divide into seven parts in such a way that'each of the three unbounded nonadjacent
regions bounded by three faces contains one and only one of the sets M3 in its interior. In the rest of the treatment we assume that the translates P + p1 separate the sets M,, ML, M3 this way. The fact that the from cony (M1 U Mk) implies, in terms of a cyclic plane P1 + p, separates notation of indices, that M1 , M2,
(24)
— (0,
0) C [(mt C) fl (mt Ck)] U (mt C1).
In terms of the notation of Definition 4, condition (24) implies that the line into two relatively open half L1 is divided by the origin (0, 0) of (a = 1,2; i 1,2,3) such that rays
RrcCrnCrnCr,
(25)
j, the vertical axis
i
D1, D2, D3 determine a three-dimensional subspace L1c (26)
Choose a fixed point p e Es plane E, + P, namely,
0) and the three lines E2)
of
E2 so that the product space of
E3).
Also
(i = 1,2,3). with the
(27) E3) parallel to E5). To prove (20) for the sets is a hyperplane in K. with E2 replaced by E3 in Definition 5, consider the sets
K1 n K2n
K1n
+P)
(i,j1,2,3,i*j). Let (a=1,2), where
is defined in Definition 4.
We will prove that
in (28) is a cell (homotopic to zero in itself). The
F. A. VALENTINE
lies between the two disjoint unbounded convex sets mt Cj1(P) n E3 + and mt C.8(p). The translates L, + p and Rf ÷ p of L, and Rf, respectively, E3). Since Cf are cones, conditions E2 + p) of lie in the hyperplane set
(25) imply that the one-dimensional sets n mt n mt Nf mt
n (R? + P)
are half-lines.
Since
each of the unbounded nonadjacent regions determined by P, +
P,
2, 3) which is bounded by three faces contains one and only one of the U u N) is intersected by the vertical ray sets M1, M,, M3, the cony (i = 1,
[(z,p): z > 0].
Hence we may choose points
e Nf sufficiently far out on Nf so that cony U U xb and cony in parallel planes, and so that cony (xi' U x' U x) c mt Cf U mt Cf U mt Cf. conv(xr U
xu
intCf n intCf n
U
U
lie
intCf *0.
The following set U U U U U Q cony U is an octahedron each of whose faces is a triangle, and the faces cony and cony U U U are parallel. Moreover, we may choose x' so that the faces are congruent and parallel. The condition x? e Nf (see (31)) and the convexity of mt Cf imply that (33)
cony (x? U
c mt C(p)
u
where i,j = 1,2,3,1 * j; a, = 1,2, a *
Hence the set K13 given by (28) is in the interior of the octahedron as given by (33). Therefore K122 is compact, can be since it is closed and bounded. Incidentally the compactness of shown in another way very easily. We will demonstrate that K123 is a cell (houiotopic to zero). In the simplest case, when C11(p) n Cr(p) = 0 (i = 1, 2, 3), K123 is a cell, having six curved
faces and eight vertices, so that it is topologically like a cube as far as the incidence of faces, edges arid vertices go. However, our proof will also cover the special cases in which the cube-like character of K123 assumes a simple but degenerate form. are disjoint open convex sets in To prove this, since mt and mt E2 + p) which separates mt C,1(p) E, + there exists a plane H, in and mt C(P). Moreover, the form of Q in (33), conditions (32) and (34), imply that H1 n H2 n H8
where
the planes
is a line in
E2 + /) n Q,
The space E + P) is divided by E2 + (i = 1,2,3) into eight octants, so that we may regard the planes
THE DUAL CONE AND HELLY TYPE THEOREMS
+ having q as a common point. Since as coordinate planes in C(p))u ri mt C7(p) = 0, i * j, and since bd Qc L, ri mt Ct(/') = 0, (mt C,'(p))], we have Q ri L•, n bdC (35)
i*j *k.
Consider, for example, the octant A containing the points course, q. Let the three faces of the octant A be denoted by H1, ,
and, of H13, H23
so that The following sets J112 n
(36)
,
H,., ri KI,,. ,
1112
ri
are each cells. For instance, H33 n K323 is bounded by either a simple closed curve consisting of the segments an arc in the bd C and an arc of the Ext
or it may be bounded by two such arcs and a segment, or it may
reduce to a segment or to a point. In any case, it is a cell. The intersection
of the octant A with the set K323 can be decomposed into a set of parallel directed segments and points with initial endpoint on one of the cells in (36) and parallel to the half-line N,1 given in (31), since c mt C(p) Hence the intersection of the octant A with K123 is also a cell in E, + P). Since this holds for the intersection of each of the eight octants, determined (i = 1, 2, 3), with K133 it follows immediately that K12, is itself a cell. by There are other ways to prove that K12, is a cell, and the reader may wish to form the proof in the same manner that was used in Theorem 14. Finally, let S be an infinite right solid cylinder in E, + P) having the vertical line p) as its axis of symmetry, and a radius such that all of
the sets (28) are contained in S for each triple of sets M1, M3, M, in the finite family sets
c 5. It is a relatively simple matter to verify that the
Sn K, n K, n (W, E, +P) Sri +p) are cells for every M and M, in Since K1 n K, n Kb n (0, 0) for i j k, the hypothesis in our theorem implies
E,) =
(37)
for every four members M, M,, in Hence, this, together with the fact that the sets (28), (36.1) are cells, implies, by Helly's generalized theorem on intersecting cells [8; 13], that
fl holds. Since (38) holds for each finite subcollection
implies that
of 5, compactness
488
F. A. VALENTINE
fl.9' * (0, 0). The dual of this condition implies there exists a plane in E, which intersects This completes the proof. all the members of It is of interest to express Rademacher's and Schoenberg's simple proof of Kirchberger's theorem t291 in terms of our notation, since it is so simple. The following is Kirchberger's first theorem [18]. THEOREM 16. Let P = {p} and Q = (q} be two compact collections of points in Euclidean n-space Then P and Q can be strictly separated by a hyperplane if and only if for each subset T of n + 2 or fewer points of P U Q there exists a hyperplane H( T) which strictly separates T n P from T n Q. Psoor. We will prove the sufficiency when P and Q are finite collections. To do this let h, and denote the support functionals for p E P, q E Q, respectively, so that (39)
h,(u)=u.p, hq(u)=u.q,
qeQ.
The dual cone C, of p and the closure of the complement of the dual cone of q, denoted by c:, have the form [(z, is), z h,(u),peP] (40) [(z,u),zhq(u),qeQJ, and (39) implies these are all closed half-spaces. Let T be n + 2 points chosen (ium P U Q. The existence of a hyperplane strictly separating T n P from T n Q implies dually that a relatively open half-line with endpoint (0, 0) in exists which is in the interiors of the cones C,, for all p T n P, q e T n Q. Belly's Theorem 11 applied to the interiors of the cones (halfspaces in fact) in (40) in the (n + 1)-dimensional space &) implies there exists a relatively open half-line in with endpoint (0, 0) which is in the interiors of all the cones C, , P P, C, q e Q. Dually, this is equivalent to the existence of a hyperplane in which strictly separates P from Q. This completes the proof when P and Q are finite collections. A simple argument involving compactness can be given to prove the case when P U Q is not finite. We will leave this to the reader. The following corollary is of some interest. The reader may prefer to restate it in less picturesque style. Cq
CoRoLL4uiY 1. Let B be a finite collection of black sheep and let W be a finite collection of white sheep in If for each set T of n + 2 or fewer sheep chosen from B U W there exists a hyperplane H( T) which strictly separates the black sheep in T from the white sheep in T, then there exists, a hyperplane which strictly separates all the black sheep in B from all the white sheep in W. PROOF. This is an immediate consequence of Theorem 16 in which the collection P consists of all the points belonging to the sheep in B, and Q consists of all the points belonging to the sheep in W.
THE DUAL CONE AND HELLY TYPE THEOREMS
In Corollary 1 has an interesting dual for sets on the sphere in order to state it, we require the following definition. denote the n-dimensional surface of the unit sphere in DEFINITION 6. Let E,,) with center at (0, 0). The the (n + 1)-dimensional Euclidean space with an open The intersection of Point (1, 0) is called the north pole of half-space is called a cap. If a cap contains the point (1, 0) and if it lies on a hemisphere of 5 it is called a convex polar cap. The complement of the closure of a convex Polar cap is called a polar cocap. It should be observed that the set of half-lines having (0, 0) as a common endpoint and intersecting a given convex polar cap is a convex cone which contains the positive z-axis [(z, 0), a > 0].
of the THEOREM 17. Suppose .9- is a finite family of sets on the surface unit sphere in Es). Suppose each member of is a convex polar cap or a polar cocap, and suppose that every n + 2 or fewer members of .9- have have a point in common. a point in common. Then all of the members of (It should be observed that by Definition 6, the members of 9 are open relative. to Se.) - •, s) be the members of .9 which are polar caps, denote the members of .9 which are polar cocaps. Let C, denote the cone formed by the set of half-lines having (0, 0) as a common endpoint and intersecting cl K,. Attach a corresponding meaning to PROOF.
Let K, (1
and let K (j C}.
1,
-
1,
-
-, t)
Since
[(a, Ø):z > 0]c intC1 [(a, 0): z < 0] c mt C,'
(j
= 1, - . -, t)
let
(i1,--.,s),
z,=h,(u) =
(j = 1, ---, t) be the equations of the surfaces of the cones C, and C,', respectively, in Es).
Hence C, C,'
[(a, u): a [(a, u): z
h,(u)J, h(u)] -
The sets M, and M,' in EM, defined as follows, Md
M,'
Lx: x
h,(u), u e E,,J,
h(u),u€EM]
are compact convex sets in EM - Now let B = i = 1, - - -, s} and W = ---,t}. Let T beany set of n+2 members of Bu W. The dual of the hypothesis of Theorem 17 implies there exists a hyperplane H(T) in EM which separates T fl B from T 11 W. Hence, Corollary 1 implies a hyper-
plane H exists which separates B from W. The dual of this statement imthat a relatively open half-line in plies in EM) exists having
F. A. VALENTINE
(0, 0) as endpoint, and lying in all the cones mt (i = 1, . . ., s) and mt C,' (j = -••, I). This, in turn, implies that all the members of 5 have a point in common. This completes the proof. Kirchberger theorem [181 is included for the sake The following of completeness. Our proof reformulates that of Rademacher and Schoenberg [29) in terms of our notation. If for and Q = {q} be two compact sets in THEoREM 18. Let P = each set T of 2n + 2 or fewer points of P U Q there exists a hyperplane H(T) which separates T n P from T n Q, then there exists a hyperplane which separates P from Q (not necessarily strictly). PRooF. As in the proof of Theorem 16, let C, and c be the closed halfspaces defined in (40). By hypothesis every 2n + 2 or fewer of these closed half-spaces, for p P. q EQ. have a ray in common with endpoint 0) in E,.). By a theorem of Steinitz [37] or Dines and McCoy [41 or Gustin with endpoint (0, 0) which (101 this implies the existence of a ray in
lies in all the half-spaces C,, p P and Ce', q Q. The dual of this implies that a hyperplane exists in E% which separates P from Q.
E) and the dual cone 4. ConcludIng remarb. The extended space often enable one to obtain an insight into the situation. The dual cones usually center the difficulties in the neighborhoods of the origin (0, 0) of R). The story is still an incomplete one since there remain theorems whose duals have not been studied. To mention one, the duals of the theorems of Helly type on the sphere, and the recent result of Grunbaum [9] on "the dimension of intersecting sets" have a close connection. The contents of Grilnbaum's paper [91 includes the following new result. THEOREM 19.
Suppose
is a finite family of convex sets in Euclidean n-
space E1. If the intersection of each n + d or fewer members of
contains a convex
set of deficiency d, where d is an integer with 0 1 should follow the same lines, and it is quite clear that an appropriate theorem of Carathéodory type will yield the
F. A. VALENTINE
492
desired result. This should be done. Also Theorem 19 and the theorems of Robinson 132] on "spherical theorems of Helly type" have an intimate connection. This is a matter which invites further investigation. The lunar dcones will play an important role. In conclusion, since it is a rare coincidence for the proofs of a theorem and its dual to be of equal difficulty, there is a double reason to investigate the dual. One may gain either a simpler proof or a less obvious theorem. 1. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Springer, Berlin, 1934. 2. C. Carathéodory, tJber den Variabiiitifitsbereich der Koeffizienten von Potentreihen, die gegebene Werte nicht annahnten, Math. Ann. 64 (1907), 95-115. 3. J. Dieudonné, Sur théorè',ne de Hahn-Banach, Rev. Sci. 79 (1941), 642-643. 4. L. L. Dines and N. H. McCoy, On linear inequalities, Trans. Roy. Soc. Canada Sect. LII 27 (1933), 37-70. 5. W. Fenchel, Convex cones, sets and
functions, Mimeographed lecture notes, Prince-
ton, 1953. (Also see Bonnesen.) 6. D. Gale, Convex polyhedral cones and linear inequalities, and M. Gerstenhaber, Theory of convex polyhedral cones, Activity Analysis of Production, and Allocation, Wiley,
New York, 1951. 7. Branko Grünbaum, On a theorem of L. A. Santalo, Pacific. J. Math. 5
(1955),
351-359.
8.
,
Common
transversals for families of sets, J. London Math. Soc. 35
(1960), 408-416.
9.
,
The
dimension of intersections of convex sets, Pacific J. Math.
12 (1962), 197-202. 10. William Gustin, On the interwr of the convex hull of a Euclidean Set, Bull. Amer. Math. Soc. 53 (1947), 299-301. 11. Olof Hanner and Hans Radstrom, A generalization of a theorem of Fenchel, Proc. Amer. Math. Soc. 2 (1951), 589-593. 12. E. Helly, Uber Men gen konvexer Kor per mit gemeinschaftlicher Punkten, Jber. Deutsch. Math. Verein. 32 (1923), 175-176. 13. , Uber Systeme aigebschlossener Mengen mit gemeinschaftiichen Punkten, Monatsh. Math. 37 (1930), 281-302. 14. E. Hille and R. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Vol. 31, Amer. Math. Soc., Providence, R. 1., 1957. 15. A. Horn and F. A. Valentine, Some properties of L-sets in the plane, Duke
Math. J. 16 (1949), 131-140. 16. A. Horn, Some generalizations of Hefly's theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923-929. 17. S. Karlin and L. S. Shapley, Some applications of a theorem on convex functions,
Ann. of Math. (2) 52 (1950), 148-153. 18. P. Kirchberger, Uber Tschebyschefsche Annaherungs Methoden, Math. Ann. 57 (1903), 509-540.
19. V. L. Klee, Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466. , Common secants for plane convex set8, Proc. Amer. Math. Soc. 5 (1954),
20.
639-641.
21. (1951), 272-275.
,
On
certain intersection properties of convex sets, Canad. J. Math. 3
THE DUAL CONE AND HELLY TYPE THEOREMS
493
22. G. Kôthe, Topologische linear Raüme. I, Springer, Berlin, 1960. 23. Folke Lanner, On. convex bodies with at least one point in common, Kungi. Fysiogr. Salisk. i Lund Förh. 13 (1943), 41-50. 24. H. Minkowski, Theorie der konvexen Korper, insbesondere Begrundung ihrea Ober.flächenbegriffs, Ges. Abh. 2 (1911), 131-229.
25. , Volumen und Oberftäche, Math. Ann. 57 (1903), 447-495. 26. N. H. McCoy. (See Dines.) 27. Theodore S. Motzkin, Convex sets in analysis, Bull. Amer. Math. Soc. (to appear). 28. R. Phillips. (See Hille.) 29. H. Rademacher and I. J. Schoenberg, Hefly's theorems on convex domains and Tchebycheff's approximation problem, Canad. J. Math. 2 (1950), 245-256. 30. Hans Radstrom. (See Hanner.) 31. M. Riesa and F. Riesz. (See comment preceding Theorem 10 of this paper.) Also see Sandgren's paper, p. 23. 32. C. V. Robinson, Spherical theorems of Helly type and congruence indices of spherical caps, Amer. J. Math. 64 (1942), 260-272. 33. Len'iart Sandgren, On convex cones, Math. Scand. 2 (1954), 19-28. 34. Richard De Santis, A generalization of HeUy's theorem, Proc. Amer. Math. Soc. 8 (1957), 336-340.
35. 1. J. Schoenberg. (See Rademacher.) 36. L. S. Shapley. (See Karlin.) 37. E. Steinitz, Bedingt konvergente Reichen und konvexe Systems. I-Il.!!!, J. Reine Angew. Math. 143 (1913), 128-175; 144 (1914), 1-40; 146 (1916), 1-52. 38. F. A. Valentine. (See Horn.) 39. P. Vincensini, Figures convexes et variétés linéaires de l'es paces euclidean a n. dimensions, Bull. Sci. Math. 59 (1935), 163-174. UNIVERSITY OF CALIFORNIA, Los ANGELES
UNSOLVED PROBLEMS Like elementary number theory, the subject of convexity lends itself readily
to the statement of interesting unsolved problems. Many of these can be appreciated on an intuitive level and may be accessible to anyone with a bright idea, for the subject (on the whole) is one of many ideas and specific approaches but little machinery. The discussion of unsolved problems was an important part of the Symposium, both informally and in a special session
devoted to them; several of the papers published here originated in such discussion. Unsolved problems are found in many of the papers, and the list below contains other problems stated during the Symposium or sent later to the editor.
V.K. W. CHENEY
M a linear subspace of R', and denote the pth-power norm in Let II for each x e let ir,x be the (unique) point of M which is nearest to x with (M, x fixed) as respect to II I,. What can be said about the behavior of
H. S. M. COXETER
If the edges of a convex polyhedron all touch a sphere of unit radius so as to form a "crate" from which the sphere cannot escape, prove or disprove that their total length is at least 9 i/T. (Cf. [1] for discussion of the related problem in which the requirement of touching is omitted.) [1J A. S. Besicovitch, A cage to hold a unit-sphere, these Proceedings, pp. 19-20. L. DANZER
Given a convex body (i.e., a convex, compact point-set with nonempty interior)
C in R'. Say its (euclidean) width (minimal distance between two parallel supporting hyperplanes) is d(C). Define its k-dimensional width dk(C) to be the maximal width attained by any intersection of C with a k-dimensional flat. Clearly d1(C) = diam (C) = d(C). I ask for the numbers q(k; n): = inf {dk(C)/d(C): C a convex body in R" (1
iii particular, for q(2; 3).
It is trivial that 495
k
UNSOLVED PROBLEMS
496
q(k+ 1;n+
but is it true, that also q(k; n + 1) Note that q(2; 3) < 1, as shown in [1).
q(k;
n)?
The example there could be simplified
(using a regular tetrahedron instead of a cube), but certainly that method is not good to prove q(2; 3) < .995 nor will it yield lower bounds for q(k; n). Of course one may ask the same questions for any Minkowskian metric over R' instead of the euclidean one. [I] L. Danzer, Uber die maximale Dicke der ebenen Schnitte eines konvexen Körpers, Arch. Math. 8 (1957), 314-316.
C. DAVIS
In n-space (n > 2) it is natural to consider, along with the diameter D and width d of a convex body K, intermediate measures. In particular, let Dk(K) = mm D(PkK), where Pk means projection to an (n — k + 1)-flat and the minimum is over all Pk; and d4(K) = max where means section by an 1flat and the maximum is over all For ellipsoids, one proves from the Fischer-Courant principle that = dk, the length of the kth principal axis. In general, of course, = D1 D and = d1 = d; however, the extension of the ellipsoid case can not go far, even for centrally symmetric K. PROBLEM. In 3-space, find the possible range of variation of d2/D2. Perhaps even the dependence of this range on dID could be found; see Besicovitch's problem. It is clear that d2/D2 1, and it is equal to 1 not only for ellipsoids but for a variety of other bodies including all those with rotational symmetry. It is equal to i/)374 for the regular tetrahedron. The smallest value I know is L/T/2, attained for a class of centrally symmetric octahedra including the regular, and including others of arbitrarily small dID. For n > 3 other possibilities arise: let Dk1k2(K) = max provided n; similarly k1 + k, I do not claim to see hope of provDkikaka, --S. ing anything about these compound quantities for general K. (Cf. Danzer's problem.) A. DVORETZKY
For a Minkowski space E
and
vi(E) = max
uIx1U=i
mm
positive integer k, ii
± x1 ±
± ...
define
±
)
j,
the minimum is over all 2k possible choices of + and — signs and the maximum is over all k-tuples x1, •- •, of unit vectors. What can be said about the numbers vk(E)? where
UNSOLVED PROBLEMS
K. FAN PROBLEM.
Are the following two rotundity properties of a Banach space
X equivalent? (1) Every sequence (2) Every sequence
in X with in X with
U (x,, +
U
+
= =
1
1
is convergent. and having no
weak cluster point of norm
0
and with center at a distance > r from
the origin, answers the question affirmatively.) II. Denote by A an algebra over R, provided with a locally convex vector space topology under which multiplication is separately continuous. A spectral element of A is an element contained in a subalgebra which is the continuous homomorphic image of some (X compact). Is a subalgebra of A, consisting entirely of spectral elements, necessarily commutative? (This question arises in connection with Part II, §3 of [1].) [1] Helmut H. Schaefer, Convex cones and spectral theory, these Proceedings, pp. 451-471.
F. A. VALENTINE—E. G. STRAUS
Does there exist a nonempty compact set S in R' such that 2 m(x) for all x e S, where m(x) is the number of convex subsets of S which are maximal relative to being convex, including x, and having dimension
n
— 1?
INDEX OF UNSOLVED PROBLEMS Besicovitch, A. S. 15, 19 Bishop, E. and Phelps, R. R. 34 Corson, H. and Klee, V. L. 37, 38, 42, 48, 49 Coxeter, H. S. M. 60, 61, 67 Danzer, L. W. 100 Danzer, L. W., Grünbaum, B. and Klee, V. L. 144—146, 148-155, 159—161, 163
Davis, C. 181, 185, 197, 198 Dvoretzky, A. 209 Grtinbaum, B. 235-264, 271-281, 290 Hammer, P. C. 302, 314, 315 Klee, V. L. 349, 352-354, 357 Phelps, R. R. 393 Valentine, F. A. 490—492 Unsolved Problems 495-500
501
109, 117—120, 122-132, 137—142,
AUTHOR INDEX Italic numbers refer to pages on which a complete reference to a work by the author is given.
Roman numbers refer to pages on which a reference is made to a work of the author. For example, under Minkowski would be the page on which a statement like the following occurs: "This theorem was proved earlier by Minkowski 17, § 2] in the following
manner... Boldface nuabera indicate the first page of the articles in this volume. Abe, Y., 116, 16* Ahiezer, N. 1., 82, 93 Alaoglu, L., 74, 93 Alexandroff, p., 125, 168 Allen, J. E., 160, 183 Amemiya, 1., 75, 83, 98 Anderson, K. W., 82-84, 86, 88, 90, 91, 93 Anderson, R. D., 21, 28, 273, 281 Ando, T., 75, 83, 86-88, 91, 92, 98
Besicovitch, A. S., 13, 15, 19, 10, 21, 24, 99, 100, 141, 150, 164, 233, 242, 254-256, 265
Bessaga, C., 37, 45, 50 Bieberbach, L., 257, 165 Bielecki, A., 152, 164, 237, 254, 265 Bing, R. H., 161, 164 Birch, B. J., 117, 164, 247, 252, 265 Birkhoff, G., 74, 93, 215, 219 Bishop, E., 27, 35, '92, 94, 401 Blanc, B., 160, 164 Blaschke, W., 141, 164, 235, 244, 251, 258,
Arms, R. J., 98 Aronszajn, N., 143, 161, 183, 200, 355-357, 858
262, 285
Ascoli, G., 94 Asplund, E., 152, 168, 235, 242, 248, 253,
Blichfeldt, H. F., 60, 70 Blumenthal, L. M. 114, 132, 138, 139, 158,
258, 264
159, 161, 164
Atsuji, M., 48, 49, 50 Auerbach, H. 261, 264
Boas, R. P., Jr., 84, 94 Bochner, S., 84, 94 Bohm, J., 70 Bohme, W., 242, 243, 185 Bohnenblust, H. F., 50, 75. 94, 108, 115,
Bagemihi, F., 151, 163 Balinski, M. L., 287, 290
Banach, S., 49, 50, 76, 85, 94, 153, 154,
123, 134, 165, 281, 303, 355, 858 Boland, J. C., 104, 127, 172 Boltyanskil, V. G., 104, 114, 115, 132, 137,
168, 450
Barlow, R. B., 847 Barlow, W., 54, 70 Barthel, W., 802 Bartle, R. G., 40, 50, 450 Baston, V. I. D., 151, 164
165, 177, 242, 244, 247, 249, 251, 252, 256, 170, 274, 284
Bonnesen, T., 14, 74, 94, 104, 116, 132, 165, 210, 222, 228, 241, 249, 251, 258,
Bauer, H., 143, 157, 163, 164, 211, 218, 219
265, 272, 281, 808, 316, 492 Bonnice, W., 116, 117, 185 Bonsall, F. F. 451, 471 Borel, A., 126, 165 Borsuk, K., 126. 137. 165, 271, 279, 280,
Seale, E. M. L., 326, 327 Behrend, F., 139, 184, 241, 164 Belgodère, p., 802 Bellman, R., 1, .11 Bendat, J., 191, 200 Bender, C., 54, 70 Benzer, S., 127, 164 Serge, C., 123, 156, 164, 354, 858 Berkes, 3., 251, 265 Berstein, J., 164
281
R. C., 241, 249, 265 Bourbaki, N., 35, 94, 215, 219, 349, 358, Bose,
461, 471
Bredon, G., 248, 264 Bremermann, H. 3., 162, 163, 165 503
AUTHOR INDEX
Bruckner. M., 290 de Bruijn, N. G., 280, 281 Brunn, H., 116, 130, 165 Buck, E. F., 241, 260, 265 Buck, R. C., 94, 241, 260, 265 Bunt, L. H. N., 117, 165 Busemann, H., 73, 92, 93, 94, 161, 165, 303 Calm, A. S., 817 Carathéodory, C., 115, 118, 151, 165, 225, 282, 279, 881, 285, 290, 486, 476, 492
Charnes, A., 327 Choquet, G., 143, 165, 168 Clarkson, J. A., 50, 74, 77, 84, 85, 94 Comfort, W. W., 133, 186 Cooper, W. W., 327 Corson, H. H., 37, 351, 353, 354, 358
EeHs, J., Jr., 73, 94 Efimov, N. V., 81, 92, 95 V. A., 46, 50 Eggkston, H. G., 13, 14, 20, 104, 108, 114, 115, 132, 135, 141, 150, 166, 233, 235, 142, 252, 255, 256, 258, 260, 266, 272-274, 276, 881, 808 Ehrhart, E., 132, 167, 247—249, 251, 252, 255, 266
Ellis, J. W., 156, 167 Erdos, P., 104, 119, 137, 149, 167, 169, 278, 280, 281, 282
Estermann, T., 237, 247, 256, 258, 266 Ewald, G., 73, 95 Fan, K., 1, 11, 77-19, 82-84, 86, 87, 91, 92, 95, 156, 167, 211
Coxeter, H. S. M., 19, 20,53,53,54,58,60-62, 64, 66, 69. 70, 71, 149, 150, 166, 225, 232
Fáry, 1., 156, 167, 239, 242, 254, 256, 257,
Croft, H. T., 278, 281 Cudia, D. F., 73, 74, 90, 94 Czipszer, J., 153, 154, 166
Fejes Tóth, L., 54, 62, 69, 70, 71, 104, 128, 144, 149, 167, 169, 263, 866, 275, 282 Fenchel, W., 14, 74, 94, 104, 116, 117, 132,
Daleckil, Yu. L., 190, 191, 200 Danzer, L. W., 99, 101, 131. 139. 144, 147, 149, 161, 162, 166, 241, 247, 252, 285, 272, 275-278, 281, 349, 358
Davenport, H., 61, 71 David, N., 303 Davis, C., 181, 187, 200 Day, M. M., 74-76, 78, 82-89, 92, 94, 210, 219, 258, 265, 352, 356, 358, 401, 450 Debrunner, H., 104, 114, 122-124, 126, 127, 129-132, 144, 152, 169, 277, 279, 280, 282 Derman, C., 3*7 De Santis, R., 119, 166, 481, 492 Dieudonné, J., 352, 353k 358, 450, 473, 492
Dines, L. L., 116, 166, 490, 492 Diaghas, A., 30* Dirichiet, G. L., 54, 71 Dixmier, J., 91, 94 Drandell, M., 156, 166 Dresher, M., 130, 166 Dudley, U. W., 486 Dukor, 1. G., 109, 166 Dunford, N., 35 Dvoretzky, A., 109, 166, 203, 210, 263, 265
Eberlein, W. A., 352, 358, 450 Edeistein, M., 46, 50, 114, 166
*66
158, 159, 165, 167, 110, 222, 223, 240, 241, 249, 251, 258, 285, 272, 181, 808, 316, 349, 359, 473, 475, 492
Few, L., 149, 166 Firey, W. 3., 244, 266 Fisher, E., 436 Floyd, B. B., 353, 859 Foguel, S. R., 90, 95
Fortet, R., 95 Franklin, S. P., 160, 167 Fréchet, M., •49, 50, 827
Frink, 0., Jr., 215, 219 Frucht, R., 129, 167 Fullerton, R. B., 358, 859, 458 Fulton, C. M., 236, 237, 266 Funk, P., 251, 262, 266 Gaddum, J. W., 159, 160, 167, 170, 3*7 Gale, D., 11, 116, 118, 120, 136, 137, 151, 154, 167, 221, 225, 2*2, 244, 266, 274, 275,
182, 285, 290, 350, 859, 490, 492 Ganapathi, P., 866 Gantmacher, F. R., 11, 340, 347 Gehér, L., 153, 154, 166, 355, 359 Gericke, H., 141, 168, 239, 262, 268, 303 Gerriets, C. 3., 264 Gerstenhaber, M., 490, 492 Ghika, A., 156, 162, 168
AUTHOR INDEX
Ghouila-Houri, A., 123, 168, 354, 359 Gillespie, D. C., 437, 450 Glicksberg, L, 77-79,82-84, 86,87, 91, 92, 95 Godbersen, C., 258, 266 Gohberg, 1. C., 137, 168 Goldman, A. J., 327 Coodner, D. B., 357, .159 Gordon, H., 133, 166
Gorin, E. A., 48, 50 Graves, I.. M., 40, 50 Green, J. W., 160, 168 Groemer, H., 150, 168, 244, 258, 267 de Groot, J., 159, 168 Gross, W., 237, 267 Grosswald, E., 253, 264 Grothendieck, A., 357, 358, 859, 450 Grotzsch, H., 152, 168 Grünbaum, B., 101, 115, 118, 120, 123, 124, 126, 129—133, 135, 137, 140, 142-144, 146, 147, 149, 150, 152-154, 158, 159, 168, 168, 168, 169, 210, 233, 236, 240, 242-244, 247254, 260, 264-267, 271, 274-279, 281, 282, 285, 290, SOS, 349, 355-357, 358, 359, 482,
490, 491, 492
Guinand, A. P., 56, 57, 71 Günter, S., 54, 71 Gustin, W., 114. 116, 132, 160, 168, 169, 492
Hadwiger, H., 104, 109, 114, 122424, 126, 127, 129-132,
137, 144, 149,
150, 152,
155, 159, 169, 237, 267, 272, 275, 277-280, 282
Hajós, G., 61, 71
Halberg, C. J. A., Jr., 150, 170 Hales, S., 53, 71 Hall, M., 816 Hammer. P. C., 156, 170, 247, 251, 253, 254, 264, 267, 291, 291, 808, 305, 315, 316, 350, 351, 859 128, 135, 142, 170, 277, 282, 357, 358, 859, 479, 480, 492
Hanner, 0., 117,
Hare, W. R. Jr., 159, 160, 170 Harrop, R., 129, 131, 170, 354, 859 Helly, E., 101, 101, 102, 103, 106, 109, 124— 126, 160, 170, 478, 492
Henriksen, M., 357, 859 Heppes, A., 137, 170, 274, 278, 282 Hubert, D., 170 Hille, E., 473, 492 Hirakawa, .1., 262, 267
505
Hirsch, W. M., 327 Hirschfeld, R. A., 95 Hirschman, 1. 1., 347
Hjelmslev, J., 116, 170 Hlawka, E., 150, 170 Hoffman, A. J., 317, 327 Hurwitz, W. A., 437, 450 Hopf, E., 170 Hopf, H., 125, 163, 279, 283 Horn, A., 121, 158, 159, 170, 479, 492
Isbell, J., 356, 359 Jacobs, W., 326, 827 James, R. C., 32, 85, 49, 50, 85, 86, 92, 93, 95, 353, 859
Jaworowski, J. W., 280, 283 Jerison, M., 211, 218, 219 John, F., 140, 170, 241, 248, 258, 267 Johnson, S. M., .127 Jung, H. W. E., 114, 132, 158, 170, 276, 283 Jussila, 0. K., 170 0 Juza, M., 355, 359
Kadec, M. 1., 45, 46, 49, 50, 86, 87. 95 Kadison, R. V., 75, 95 Kakutani, S., 95, 133, 170, 308 Kalisch, G. K., 161, 171 Karlin, S.. 75, 94, 104. 108, 114, 115, 120, 121, 123, 158, 165, 171, 329, 847, 350, 359, 451, 459, 471, 479, 492
Kato, T., 200 Keller, 0. H., 37, 50 Kelley, J. L., 143, 171, 215, 219, 357, 360 Kelly, L. M., 73, 95, 138, 139, 171 Kelly, P. J., 242, 267, 303 Kendall, D. G., 143, 171 Kepler, J., 53, 71 Kijne, D., 129, 171 Kirchberger P., 114, 116, 171, 475, 488, 490, 492
Kirszbraun, M. D., 153, 154. 171 Klee, V. L., 21, 23, 24, 25, 27, 35, 37, 37, 45, 50, 74, 79, 83, 85, 87-90, 92, 93, 95, 101, 104, 106, 114, 116, 117, 120, 121, 123, 124, 131, 133, 156, 165, 167, 169, 171, 247, 252, 265, 267, 273, 276-278, 280, 281, 283, 305, 307, 308, 315, 349, 349-353, 858-360,
393. 401, 473, 479, 482, 492
AUTHOR INDEX
506
Klein, M., 827 Knaster, B., 123, 171, 275, 280. 283 Kneser, H., 156, 171 Kneser, M., 155, 171 Knothe, H., 237, 267 Konig, D., 103, 106, 171 Konstantinesku, F., 92, 96 Korányi, A., 188, 200, 201 A., 156, 171, 237, 249, 254, 267 Kothe, G., 353, 360, 438, 450, 475, 498 Kovetz, Y., 264 Kozinec, B. N., 255, 267 S. N., 82, 96 Krasnosselsky,
M. A., 109, 114, 171, 172
Kraus, F., 200 M. G., 82, 92, 93, 219, 340, 347, 451, 471
Krzyz, J., 347 Kubota, T., 116, 163, 238, 268 Kuhn, H. W., 11, 156, 172 Kuiper, N. H., 129, 131, ]59, 172 Kuratowski, C., 123, 171, 285, 288, 289, 290 Lagrange, R., 132, 172
Land, A. H., 327 Landau, 454 Lanner, F., 108, 172, 478, 498
Laugwitz, D., 73, 96,
139,
166, 241, 242,
280, 283 Macheath, A. M., 236, 237, 268 MacCoil, L. A., 304
MacDuffee, C. C., 200 Mairhuber, 5. C., 115, 172 Marchaud, A., 159, 172 Marcus, M., 200 Markus, A. S., 137, 168 Matumura, S., 261, 268 Mayer, A. E., 160, 172 Mazur, S., 42, 51, 82, 96 Mazurkiewicz, S., 123, 171 McCoy, N. H., 116, 166, 490, 492
McMinn, T. J., 24, 25 McShane, E. 5., 96, 153, 154, 172 Meizak, Z. A., 132, 172 Menger, K., 138, 160, 172 Meschkowski, H., 61, 71 Michaei, E., 161, 172, 353, 860 Mickle, E. 5., 153, 154, 172 Milman, D. p., 90, 96, 211, 218, 219 Mimes, M. W., 75, 96 Milnor, 5., 129, 178 Minkowski, H., 60, 71, 74, 98, 115, 147, 178, 233, 246, 247, 268,
265, 268, 303
Lavrent'ev, M. A., 25!, 254, 268 Leech, J., 54, 71 Leichtweiss, K., 135, 141, 172, 241, 247, 248, 258, 268, 276, 283, 303
Lelek, A., 161, 172 Lekkerkerker, C. G., 104, 127, 172 Lenz, H., 139, 166, 241, 258, 265, 268, 272, 275, 276, 283 Leray, 5., 126, 172
Levi, F. W., 109, 123, 137, 138, 159, 172, 236, 237, 255, 256, 288,
Lusternik, L. See Lyusternik, L. A. Luxemburg, W. A. J., 75, 98 Lyusternik, L. A., 251, 254, 268, 271, 279,
277,
278, 283
Levin, E., 150, 170
Lighthall, H., Jr., 827 Lindenstrauss, J., 355, 357, 358, 360 Linis, V., 256, 268 Lippmann, H., 304
Loewner, C., 200 Long, R. G., 51
Lorch, E. R., 73, 96 Lovaglia, A. R., 77, 82, 83, 86, 90, 92. 96 Lowner, K. See Loewner, C.
304, 493
Minty, G. 5., 154, 173 Miyatake, 0., 262, 268 Molnár, 3., 114, 125, 158, 173 Monge, G., 317, 327 Monna, A. F., 156, 173 Morton, G., 827 Moser, L. 280, 283 Moser, W., 280, 288
Motzkin, 'F. S., 115, 118, 124, 151, 156, 158-160, 169, 173, 247, 268, 279, 282, 285, 285, 290, 305, 350, 360, 361, 389, 392, 436, 477, 493
Nachbin, L., 135, 142, 143, 173, 277 283, 355-357, 360
Nakamura, M., 201 Nakano, H., 75, 76, 78, 82, 91, 96 Nasu, V., 304 Nef, W., 156, 178 Neumann, B. H., 115, 178, 247, 249-252, 268
AUTHOR INDEX
Newman, D. J., 242, 251, 252, 268 Nijenhuis, A., 161, 173 Nitka, W., 161, 172 Nohi, W., 264 Nordlander, G., 96 Ohmann, D., 804 Ostrowski, A., 201
j. F., 240, 268, 272, 274, 288 Panitchpakdi, P., 143, 161, 168, 355-357,
P11,
358
Pasqualini, L., 160, 173 Pauc, C., 161, 174 A., 45, 49, 50, 96 Perkal, J., 272, 273, 283
Pettis, B. J., 90, 96 Petty, C. M., 237, 268, 308, 304 Phelps, R. R., 27, 35, 81, 86, 90, 92, 93,
Rényi, C., 151, 175 Yu. G., 242, 269, 304 Révész, P., 278, 282 Riesz, F., 97, 478, 498 Riesz, M., 478, 498
Ringrose, J. R., 92, 97 Rio, S. T., 305, 316 Robinson, C. V., 116, 138, 158, 175, 492, 498 Rogers, C. A., 60, 62. 70, 71, 137, 143, 147, 149, 166, 167, 175, 210, 257, 258, 269 Rolewicz, S., 210, 210 Roy, S. N., 240, 265
Ruben, H., 56, 71, 347 Rudin, M. E., 353, 360 Rund, 1-1., 73, 97
Ruston, A. F., 82, 97 Rutman, M. A., 211, 218, 219, 451, 471 Rutovitz, D., 260, 269
94, 96, 393, 401 Phillips, R.. 473, 492
Salkowski, E., 262; 269
Pleijel, A., 269 Plunkett, R. L, 161, 174, 357, 860 Pólya, G., 264, 333, 847 Poritsky, H., 403, 486 Poulsen, E. 'F., 155, 174 Prager, W., 326, 327 Prenowitz, W., 156, 174 Proschan, F., 347 Proskuryakov, I. V., 109, 174 Ptlk, V., 115, 174, 353, 360, 437, 450 Pucci, C., 242, 269
Sandgren, L., 108, 109, 120, 175, 349, 860, 473, 475, 477, 479, 490, 493 Santalô, L. A., 114, 121, 124, 129, 130, 133, 138, 158, 160, 175, 272, 283 Sasaki, M., 75, 83, 93 Schaefer, H. H., 451, 471 Schläfti, L., 54, 57, 58, 71 L. G. Schnirelmann, L. See Schoenberg, 1. J., 108, 114-116, 130, 154, 174, 175, 347, 486, 475, 488, 490, 493 Schopp, J., 144, 175, 251, 269, 277, 283 Schur, 1., 201 Schutte, K., 54, 71 Schwartz, J., 35 Schweppe, E. .1., 159, 175
Rabin, M.,
109, 174
Rademacher, H., 108, 114-116. 130, 174, 258, 269, 475, 488, 490, 493 120, 129, 131, Rado, R., 109, 115, 117, 248, 252, 150, 152, 156, 160, 170, 269, 350, 360 Radon, J., 103, 107, 115, 147, 159, 174, 246, 247, 262, 264, 269 RSdstrom, H., 117, 157, 162, 170, 174, 479, 480, 492
Radziszewski, K., 237, 254, 265, 269 Ramsey, F. P., 350, 360 Rankin, R. A., 61, 70, 71, 149, 174 Rebassoo, H. L., 291 Rédei, L., 239, 254, 257, 266 Rémès, E., 115, 174 Rényi, A., 151, 175
Samuel,
P, 215, 219
I. E., 73, 97 Selfridge, J. L., 275, 283 Semadeni, Z., 356, 359 Shapley, L. S., 108, 114, 115, 120, 121, 123, 158, 165, 171, 350, 359, 479, 492
Shephard, G. C., 143, 147, 175, 236, 257, 258, 269
Sherman, S., 191, 194. 200 Shimogaki, T., 92, 97 Shimrat, M., 114, 175 Shkliarsky, D., 280, 284 Sholander, M., 260, 269 Sierpiñski, W., 51, 355, 360 Skornyakov, L. A., 156, 175
AUTHOR INDEX
508
Smith, T. J., 295, 297 Smith, W. E., 827 Smulian, V. L., 75, 77-79, 83, 89-91, 97, 352, 860, 450
Taylor, A. E., 90, 97, 258, 270 Tietze, H., 150, 176 Toeplitz, 0., 436 Tucker, A. W., 11, 827
Snirel'man, L. G., 114, 115, 175, 271, 279, Umegaki, H., 201 Ungar, P., 181, 251, 262, 270
280, 283
Sobczyk, A., 247, 267, 291, 303 Sokolowky, D., 827 Solian, P. S., 137, 175
Valentine, F. A., 104, 108, 109, 114, 120,
Soos, G., 176
121, 130, 153, 154, 156, 160, 170, 176, 177,
Ste&in,
304, 349, 860, 473, 479, 492
S. B., 81, 92, 95
Steenrod, N., 129, 176 Stem, S. K., 233, 236, 968,
237,
249, 254,
255,
269
Steiner, J., 238, 269
Steinhagen, P., 141, 152, 176, 272, 284 Steinhaus, H., 249, 250, 261, 269 Steinitz, E., 115, 159, 176, 285, 290, 490, 491, 493 Sternbach, L., 51
Stewart, B. M., 255, 270 Stieltjes, T.-J., 160, 176 Stinespring, W. F., 201 Stoelinga, T. G. D., 117, 176 Stoker, J. J., 37, 51 Stone, M. H., 156, 178, 211, 219 Straszewicz, S., 116, 132, 176, 278, 284 Straus, E. G., 115, 150, 161, 170, 171, 178, 389
Veblen, 0., 159, 177 Verblunsky, S., 114, 132, 177 Viet, U., 136, 177, 249, 270 Vigodsky, M., 158, 177 Vincensini, P., 114, 129, 131, 159, 177, 493
Vinogradov, A. A., 82, 98 Volkov, V. 1., 135, 177, 276, 884 Voronoi, 6., 54, 71 de Vries, H., 159, 162
Wada, J., 86, 88, 89, 97 van der Waerden, B. L., 54, 71 Wahlin, G. E., 114, 132, 164 Wallinan, H., 211, 215, 219 Well, A., 126, 177, 210 Weston, J. D., 354, 359 Whitehead, 5. H. C., 177 Widder, D. V., 847
Su, B., 238, 270
Sundaresan, K., 97 Surányi, J., 151, 175 Süss, W., 136, 176, 237, 247, 251, 258, 270 S., 151, 176 Swinnerton-Dyer, H. P. F., 150, 176 Sz.-Nagy, B., 97, 135, 142, 176, 201, 248,
Yaglom, I. M., 104, 114, 115, 132, 174, 242, 244, 247, 249, 251, 252, 256, 270, 274, .684
Yamamuro, S., 75, 83, 97 Yang, C.-T., 280, 284 Yoneguchi, H., 116, 168 Young, J. W., 159, 177
270, 277, 284
Szego, G., 264, 847 Szekeres, G., 119, 167
Taft, R. G., 285, 290 Takesaki, M., 801
Zaguskin, V. L., 139, 177, 241, 270 Zalcwasser, Z., 437, 450 Zarankiewicz, K., 264 Zaustinsky, E. M., 133, 177 Zindler, K., 242, 261, 270
SUBJECT INDEX Autothety, 362 centers of, 376 Autotranslation, 362 Axioms of convexity, 109, 155-160
2-center, 363 4-hull, 363 between, 216 convex, 215 convex hull, 216 extreme. 216 supporting subset, 217
Base, 310 Basic dimension, 365 Basis, 229
substitutive, 37, 41-45 Bauer's minimum principle, 211, 218 BdS, 473 Bernoulli numbers, 57 Binary intersection property, 356 Binormals, 293, 415 Borsuk's conjecture, 137 problem, 271 Borsuk-Ulam theorem, 221, 222 Boundary points, linear combinations of,
inductive, 211 stable,
U
211
inductive, 212 stable, 211
Acentricity, 389 Additive functions, 309 semigroup, 385 Adjacent sets, 147, 149-151
389
Boundary, outer, 389 Breadth, equivalent, 296
Admissible
n-tuple, 389 side, 403 Affine center, 382, 384 Algebra locally convex, 454 spectral, 454
Cap, 489 Carathéodory's theorem, 103, 115-117, 313 Centers halfcone, 377 of endothety, 375, 376 of symmetry, 375 Central symmetry, 373 Centroid, 240 Chain, 46-48 Characteristic cone, 38-40 Chebyshev set, 81, 90, 92, 93, 401 Circle
Ambiconvex hull, 378
pair, 378 polyhedral sets, 385 sets, 378, 384 Ambiguity, 286 Analytical representation, 291 Antimonotone, 365 Antipodes, 222 Antitonic functions, 310 Approximation theory, 110-115 Arc length function, 299, 300 Asphericity, 203 Associated complex spectral measure, 464 Associated convex bodies, 292, 294, 296 Associated modular, 91 Asymmetry, measure of, 233 Asymptotic upper bound, 70 Autothetic sets, 375
a characterization of, 99 involute of, 419 Circular helix, 418 Circumscribed bodies, 112, 134-140 circle, 221 Classes complete, 366 gapless, 366 Classification, 362 Closed multiplicative semigroups, 373-375 Closure, convex, 306 509
SUBJECT INDEX
510
Closure functions, 309 Kuratowski, 310 CIS, 473 Combinatorial topology, 109, 125, 126, 129 Combinatorially equivalent 227 isomorphic, 227 Compactness, weak, 352, 353 Complemental, 362 Complementary semiflats, 378 Complete classes, 366 endovector sets, 364 families, 365 hull, 365 Compressed, 364 Complex convexity, 157, 162-163 Computation, 67 Cone
centers, 376 characteristic, 38-40 convex, 37-43, 48-50 dual, 475 generating, 456 lunar, 490 normal, 452 positive, 37, 41—43, 49, 452
support, 28 Cones, 371
Conical singular point, 23 Conics, 139 Conjugate modular, 91 Consistent, 1 Constant projection, 356
retraction, 356 width, 141 Constant breadth, 291, 293, 294 convex curves, representation of, 291 curves, 291
relative to
298
surfaces, 291
with respect to
301
Continuity, 315 properties, 379 uniform, 38, 48 Continuous, 86 functions (spaces of), 45, 46 norm, 86 Contractive functions, 309
Cony S, 473
Convergence is almost uniform, 442 Convergence is uniform in the mean, 442 Convex bodies, 37-41, 45-47, 73, 74, 221 associated, 292, 294, 296 Convex closure, 306 cone, 28, 107, 129, 158 curves, 181
curves and rectangles, 99 function, 123 hull, 103, 115-119, 159-160, 162, 225, 306, 308
polygons, 225, 230, 302 polyhedra, 383
region R, 403 set, analytic, 222 sets, 368 totally, 355 Convexity axioms of, 109, 155-160 complex, 157, 162, 163 connected, 117, 118 generalized, 155—163
metric, 161, 162 preserving properties, 342 projective, 159 spherical, 157-159 Core, 362 Cover minimal, 274
universal, 273 Covering, 111-113, 133-140, 145-150, 354 Critical point, 246 set, 246 Cross-polytope, 61 Cubic close-packing, 53 Cuboctahedron, 53
Curvature, 73, 416 centroid, 241 Curve envelope, 405
of regression, 413 osculating, 415 Curves, constant breadth, 291 Curves, convex, 181
of constant Minkowski breadth, representation of, 297 Cusps, 411
SUBJECT INDEX Deficiency, 476
Densest packing of equal spheres, 60 Density, 54
221, 291 affine, 254
Diameter,
Diametral
chords, 292, 293, 296 lines, 291, 294, 296 line family, 294, 295, 298 line family, essential, 294, 304
Differentiability of the norm, 89 Differential equivalence, 291, 294 Dimensionally ambiguous, 286 Direct product, 382, 384 Dirichlet region, 54 Distance-functions, 235 D-localized, 79, 83
Dodecahedron rhombic, 54 trapezo-rhombic, 54 Domain bounded, 311, 313 finite, 311
Essential diametral line, 296 Euclidean breadth, 292 Exact, 356 Exactness of various constants, 355 Existence set, 81, 92, 93 Expansion constant, 355 Expansion function, 309, 311, 313 Expansive function, 310 Exposed
310
140, 158, 349
Edge, 225
Element, quasi-interior, 458 Ellipsoid,
Euclidean cells, 129-139, 141, 144—
152, 155 Endo-A, 363
Endomorphism, positive, 456 Endoring,
375
Endothetic
sets, 373
Endotranslation,
363
sets, 364 Endovectors, 361 if. Envelope, 292 curve, 405 Endovector
of family of straight lines, 406 Equivalence of breadth, 292 Equivalent, 86 in
75, 82
I-
additive functions, 309 cone, 475 properties, 89 Duality, 74, 88-90, 108,
point,
Extended strength, 381 Extended topology, 305, 310, 313 Extended topological system, 310 Extension property, 142, 357 Extreme point, 82, 215, 308, 357, 358 relative, 398, 400 Extreme subset, 215 Euclidean geometry, 73 Even, 80, 83, 86 modular, 88 closure function, 310 contractive function, 310 expansive function, 310 interior function, 310 limit function, 310 limit point, 310 primitive function, 310
finite function, 311, 313 finite closure function, 312 finite expansive function, 312 Double limit condition, 448 Dual,
51L
breadth, 292, 293, 299
Fexposed point, 75, 82 hyperplane, 74 nonsupport point, 75 rotund, 75 rotundity, 75, 82 smooth, 75 support point, 75, 82 hull, 211 kernel, 211 Face, 227
Face-centered cubic lattice, 53 Family A is said to be quasi -equicontinuous, 449
Family
of lines, 297 of straight lines, 406 Families, Complete, 365 main, 365 maximal, 366 translative, 366
SUBJECT INDEX
512
Families of sets homothets, 131, 132, 134—136, 146-149, 151, 152 disjoint, 130—132, 151—153
intersectional, 123-125 translates, 131—138, 144—147, 149, 150, 155
Fan, 41, 42 Filter, 214 Finite, 83 Finsler spaces, 73 First norm, 76, 91, 92 Flatness, 83 Flats, 365 osculating, 405 Fourier series, 299 Fréchet differentiable, 78, 83, 89 Fresnet equations, 416 Full,
452
g-neigbborhoods, 310
Gallai problems, 128, 144, 152, 153 Gapless classes, 366 hull, 366 Gateaux differentiable, 78, 82, 83, 89 Gauge, 41, 42 Generalizations of convexity, 155-163 of Helly's theorem, 119-128 Generating cone, 456 Generator, single, 368 Geometry metric, 73 Minkowskian, 73 Riemannian, 73 Graphs, 118, 127, 152 Grids, 367 Group, 312
Halfcone centers, 377 Half cones, 375 Halfspace, 111, 113, 140—142, 156, 161, 162
Harmonic function, positive, 406 Helicord, involute, 418 Helix, circular, 418 Helly, Eduard, 101, 102 Helly
problems and numbers, 124, 127, 128 type theorems, 478
Helly's theorems, 101—104, 106-109
Hereditary, 212 Hermitian, 2 matriceS, 1 Herpolhode, 422 Hexagonal close-packing, 54 Hexagonal points, 242 Homothets, 105, 106 Homothety, 362 Hull, 362 ambiconvex, 378 complete, 365 gapless, 366 Hyperconvex, 356, 357 Hyperplane, 74, 226, 227 Idempotent, 309
Identical arc length functions, 299 Inclusion preserving, 309 Indecomposable polyhedra, 244 Inequalities, linear, 1 Infinite dimensional Finsler spaces, 73 Infinitely increasing, 83, 86 Inscribed circle, 221 Integration, 68 Interior functions, 309, 310 Internal associated convex bodies, 296
Internal f-primitive functions, 310 Intersection basis, 307, 308 pattern, 126, 127 theorems, 101—163
Intersectional, 362 Intv S, 473
Invariant points and sets, 238 Inverse
star centers, 377 stars, 374, 377 overstar center, 378 overstars, 374 strength, 378 Involute helicoid, 418 Involute of circle, 419 Involutes, spherical, 424 Irreducible sets, 244 subbody, 296 Isolated point, 308 Isomorphic, 84, 85 Isomorphism, 73, 85, 87, 88
513
SUBJECT INDEX
Isotonic, 309
join, 362 Jung's theorem and its relatives, 112, 113, 131-137, 140, 145, 146, 154
k rotund, 77, 82-84 Kernel, 211
Krein-Milman theorem, 211, 215, 218
Krein's theorem on the convex extension of a weakly compact set, 440 Kuratowski closure function, 310 Lattice, 160 packing, 60 Level sets, 382, 384 Limit functions, 309, 310, 313 point, 310 Lindelöf property, 351, 353, 354 Line families, 294-298 outwardly simple, 291, 296, 297, 299,302 Linear combinations of boundary points, 389 if. homeomorphisms, 73 manifolds, 312, 314 programming, 317 space, 312, 313
Linear inequalities, 1 in Hermitian matrix variables, 1 in the n variables, 403 Linear'y closed, 377 hoineomorphic, 86 Lines, 297 Lipschitzian, 221, 297 transformations, 48, 152-155
Local uniform smoothness, 73 Locally Banach, 73
convex algebra, 454 convex vector lattice, 459 uniformly rotund, .77, 82-84, 86, 87 uniformly smooth, 78, 82 Löwner's ellipsoid, 241 Lunar d-cone, 490 Main families, 365 Map, quasi-interior, 458 Matrix-convex, 187
Matrix-monotone, 187 Maximal families, 366 sets, 314 Measure of asymmetry, 233 of
24
spectral, 454 Measurement, quantum-mechanical, 197 Measures of symmetry, 233 Metric cells, 355 convexity, 161, 162 geometry, 73 projection, 93 spaces, 139, 143, 144, 153-155 Midpoint locally uniformly rotund, 77, 82-84, 88 smooth, 78, 83 Midpoint local uniform smoothness, 73 Milman-Rutman's theorem, 211, 218 Minimal base, 314
base of neighborhoods, 305 cover, 274 families, 366 intersection basis, 312, 314 neighborhood, 308 Ø.supporting subsets, 217 union basis, 312 Minkowski spaces, 73, 133-131 Minkowskian geometry, 73 metric, constant breadth, 291, 293, 294 metrics, 291, 297 Modular, 75, 79, 92 norm, 76, 87, 92 Modulared linear lattices, 85 Modulated vector lattices, 75, 79, 83, 8690, 92 Modulars, 81 Modulus of rotundity, 85 Monge sequence, 317-319, 325 Multiplicative semigroup, 364, 370 Nearideals, 385 Neighborhood, 310, 312, 314 base, 312 Neighborhoods, 305, 308 Neighborly polyhedra, 118, 151
514
SUBJECT INDEX
Neighbors, 225 Non-atomic, 83 Non-Euclidean space, 69 Non-lattice packing, 61 Non-support point, 75 Normal cone, 452 Normality, 353, 354 Norm-closed, 377 Normed linear spaces, 133, 143, 152-155
Ordered locally convex algebra, 454 Ordered topological vector space, 452 Osculating curve, 413 flats, 405 plane, 413 Osculatory, 79, 83
o. s. line family, 297 Outer boundary, 389 Outwardly simple, 297 Overstar center, 378 Overstars, 375 inverse, 374, 378
Packing, 54 Packings, 147,. 149, 150 Pair, ambiconvex, 378 Pairproduct, 378, 384 Paracompactness, 353 Parallel, 296 Parallelotopes, 127, 129-131, 137, 142-145, 150-152, 154
Pedal function, 292, 294, 297, 299, 302 Period, 367 Permutationally symmetric, 364 Plane, osculating, 413 Point locally uniformly totund, 78, 82, 83,
Polyhedra, 106, 118, 119, 129, 150-152 conveX, 383
Polyhedral graphs, 285 dimension of, 285 Polyhedron, regular, 383 Polytopes 225-227, 230 Positive cone, 37, 41-43, 49, 75, 452 definite, 1 endomorphisms, 456
functions, totally, 337 harmonic function, 406 spectral element, 465 spectral measure, 461 Positivity, total, 329 if. Primary f-limit point, 310 f-self-dense function, 310 self-dense, 314
Primitive functions, 309 Principal filters, 214 normal, 415 Product direct, 382, 384 of Banach spaces, 83 spaces, 83, 87 Projection constant, 356 operator, 76, 80 Projective convexity, 159 functions, 309 Proper, 452 ambiconvex set, 379 dimension, 372 endothety, 385
86
Point of F-smoothness, 75 Point symmetrization, 293 Points
of rotundity, 74 of smoothness, 74 Polar cap, 489 Polar body, 74 Polarity, 349 Polhode, 422 Polygons convex, 225, 230 regular, 230
Quasi-interior element, 458 map, 458 Quasipencil, 297 Quotient spaces, 83, 84 Quasiconvex, 46-48 r-neighborhood, 310, 312, 314 Radon's theorem, 103, 107-109, 117, 118, 159
Rectangles, convex, 99 Reflexive, 91
SUBJECT INDEX
Reflexivity, 91 Regression, curve of, 413 Regular 230 n-gon 600-cell, 64 polygons, 230 polyhedron, 383 Relative extreme point, 398, 400 Relatively constant breadth curves, finite constructions, 300 Representative spherical triangle, 421 Resolvent, 454 Retraction constant, 356 Reuleaux triangle, 13 Rhombic dodecahedron, 54 Riemannian geometry, 73 Ring, 312 Rolling without slipping, 422 Rotation total, 408 vector, 420 Rotund, 74, 77, 82-86, 91-93 Rotundability, 87 Rotundable, 87, 89 Rotundity, 73 if. Ruled surface, 413
Saturated, 453 Scalar product, 228 Schläfli function, 56 Schur-convex, 198 Second norm, 76, 83, 91, 92 Segment, 225 Self-dense function, 310 Semiflats, complementary, 378 Semigroup, 312 additive, 385 closed multiplicative, 375 connected multiplicative, 373, 374 multiplicative, 364, 370 Semiregularity, 91 Semiregular polyhedron, 383 Semiring, 368 Semispaces, 156, 305, 306, 314, 350, 351 Separation by hyperplanes, 106, 107, 109, 111, 130 theorem, 27 Set of constant width, 15 Sets, 365 ambiconvex, 384
515
ambiconvex polyhedral, 385 autothetic, 373, 375 Chebyshev, 401 complete endovector, 364 convex, 368 endothetic, 373 level, 382, 384 symmetric, 373 Simple, 15 orderings, 306 Simplex, 112, 113, 123, 136-138, 140, 141, 143, 151
Single generator, 368 Single integral generator, 369 Six-partite point, 241 Smooth, 74, 75, 78, 82-87, 91-93 Smoothability, 87 Smooth conjugate, 92 Smoothness, 73, 74, 85, 90, 93 Space, Minkowski, 73 Spectral algebra, 454 element, 454 element, positive, 465 Spectral measure, 454 associated complex, 464 product, 461, 463 support, 454 Spectrum, 454 Sphere, 105, 152 Spherical convexity, 157—159 424 Spherical
Spherical to within Stability, 1 Star centers, 377 Stars, 374, 377, 380 inverse, 377, 381
203
Starshaped sets, ill Steinitz's theorem, 115, 116 Stone's theorem, 211, 214 Strength, 379, 381, 389 Strict convexity, 79, 80 Strict 'B-cone, 453 Strictly convex, 80, 83, 86, 91, 222 Strictly positive linear form, 459 Strip, 292 Strong cones, 371 stars,
374, 377
halfcones, 375 overstars, 375
SUBJECT INDEX
516
Strongly ambiguous, 286 Subpencil, 300, 302 Subspaces, 83, 84 Substitutive basis, 37, 41—45 Superminimality condition, 243 Support cone, 28, 394ff. functional, 23ff., 393, 473, 475 point, 27-35, 75, 82, 393 spectral measure, 454 theorem, 27 Supporting hyperplane, 225 strips, 292-294 Suppression, 364 Surface-area centroid, 240 Surface, toroidal, 424 Surfaces, constant breadth. 291 Symmetric body, 222 convex bodies, 129, 134-137, 140, 141, 145-147, 149, 150
permutationally, 364 sets, 373 Symmetroid, 293-295 Symmetrization, central, 292, 293 Symmetry, centers of, 375 central, 373 measures of, 233 Topological classification, 37 Topology, extended, 305, 310, 313 weak, 352, 353 Toroidal surface, 424 Torsion, 416
Total rotation, 408 Totally convex, 355 Totally positive functions, 337 Translative endovector sets, 367 families, 366 Translativity, 366 Transportation problem, 317, 318 Transversals, 110, 111, 114, 121, 129-132 Trapezo-rhombic dodecahedron, 54 Tyhonov cube, 81
Ultimate root, 315
Ultra filter, 214 Umbra! notation, 56 Unambiguous, 286 Uniform
continuity, 38, 48 flatness, 83 rotundity, 74, 87 Uniformly
convex, 80, 83, 9L 92 even, 80, 83, 91, 92 Fréchet differentiable, 75, 78, 83, 89 Gateaux differentiable, 78, 83, 89 increasing, 91 rotund, 74, 77, 82-85, 87, 91 smooth, 78, 82-85, 91 Unique extension property, 81, 93 Unional, 362 families, 371 Unit ceLl, 37, 39-41, 45, 49, 50
tangent vector, 416 Univalent, 221 U'iiversal cover, 273 Universally continuous, 90
v-inverse star, 381 v-star, 380 Variation diminishing property, 336, 337 Vector
lattice locally convex, 459 rotation, 420 Virtual envelope, 297, 300 Visible, 111 Voronoi polyhedron, 54
Wallman's theorem, 211, 214 Warehouse problem, 317, 319, 322 Weak compactness, 352, 353, 437 topology, 352, 353 Weak*
locally uniformly rotund, 77, 82 uniformly rotund, 79 Weakly ambiguous, 286 k rotund, 77, 82 locally uniformly rotund, 77, 82, 83, 86
uniformly rotund, 79 Width, 140, 141, 154, 221, 222