Modern Birkhauser Classics Many of the original research and survey monographs In pure and applied mathematics published...
12 downloads
792 Views
17MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Modern Birkhauser Classics Many of the original research and survey monographs In pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being rereleased in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.
Notions of Convexity
Lars Hormander
Reprint of the 1994 Edition Birkhauser Boston • Basel • Berlin
Lars Hormander Lund University Center for Mathematical Sciences SE22100Lund Sweden
Originally published as Volume 127 in the series Progress in Mathematics
Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 00A05, 01A60, 0302, 26A51, 26B25, 3102, 31B05, 31C10, 3202, 32F05, 32F15, 32T99, 32U05, 32W05, 35A27, 52A40 Library of Congress Control Number: 2006937427 ISBN10: 0817645845 ISBN13: 9780817645847
eISBN10: 0817645853 eISBN13: 9780817645854
Printed on acidfree paper. \®
©2007 Birkhauser Boston BirkMuSCr i^ All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer SciencefBusiness Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com
(IBT)
Lars Hormander
Notions of Convexity
Birkhauser Boston • Basel • Berlin
Lars Hormander Department of Mathematics University of Lund Box 118,8221 00 Lund Sweden
Library of Congress Cataloging InPublication Data
Hormander, Lars, 1931Notions of convexity / Lars Hormander. p. cm.  (Progress in mathematics ; v. 127) Includes bibhographical references and indexes. ISBN 0817637990 (acid free). 1. Convex domains. I. Title. II. Progress in mathematics (Boston, Mass.) ; vol. 127 QA639.5.H67 1994 9432572 515.'94dc20 CIP
Printed on acidfree paper © Birkhauser Boston 1994
Birkhduser
Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0817637990 ISBN 3764337990 Typeset by the author in AMSlj^X. Printed and bound by QuinnWoodbine, Woodbine, NJ. Printed in the U.S.A. 9 8 7 6 5 4 3 2 1
PREFACE
The term convexity used to describe these lectures given at the University of Lund in 199192 should be understood in a wide sense. Only Chapters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudoconvexity (plurisubharmonicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R^, so Chapter III is devoted to subharmonic functions in R"^ for any n. Existence theorems for constant coefficient partial differential operators in R'^ are related to various kinds of convexity conditions, depending on the operator. Chapter VI gives a survey of the rather incomplete results which are known on their geometrical meaning. There are also natural classes of "convex" functions related to subgroups of the linear group, which specialize to several of the notions already mentioned. They are discussed in Chapter V. The last chapter. Chapter VII, is devoted to the conditions for solvability of microdifferential equations, which can also be considered as a branch of convexity theory. The whole chapter is an exposition of a part of the thesis of J.M. Trepreau. Thus the main purpose is to discuss notions of convexity — for functions and for sets — which occur in the theory of partial differential equations and complex analysis. However, it is impossible to resist the temptation to present a number of beautiful related topics, such as basic inequalities in analysis and isoperimetric inequalities. In fact, this gives an opportunity to show how conversely the theory of partial differential equations contributes to convexity theory. Originally I also planned to discuss the role of convexity in linear and nonlinear functional analysis, but that turned out to be impossible in the time available. Another topic which is conspicuously missing is the theory of the real and the complex MongeAmpere equations, which should have been presented in Chapters II and IV. Convexity theory has contacts with many areas of mathematics. However, only applications in complex analysis and the theory of linear partial differential equations are discussed here, without aiming for completeness. I hope that in spite of that the book will prove useful for readers with main interest in other directions, and that it does justice to the beauty of the subject. To minimize the number of references relied on I have often referred to
my books denoted by ALPDO and CASV (see the bibliography at the end) instead of original works. Further references can be found in these books. At the beginning of the notes no prerequisites are assumed beyond calculus and linear algebra. Measure and integration theory are required in Section 1.7 and from Chapter III on. Distribution theory has been used systematically from Chapter III when it simplifies or clarifies the presentation, even where it could be avoided. However, only the most elementary part of the first seven chapters in ALPDO are required. Some background in differential geometry is assumed in Section 2.3, and the proof of the FenchelAlexandrov inequality there requires some knowledge of elliptic differential operators. At the end basic Riemannian geometry is also required, and Section 6.2 assumes familiarity with pseudodifferential operators. The last section. Section 7.4, assumes some background in analytic microlocal analysis, and some knowledge of symplectic geometry is needed in Section 7.3. Only the simplest facts from functional analysis are needed except in Section 6.3 where deeper results on duality theory are used. However, these are exceptions which can be bypassed with no loss of continuity. Apart from these points the notes should be accessible to any graduate student with an interest in analysis. As already mentioned Chapter VII is based on J.M. Trepreau's thesis. The presentation here owes much to the patience with which he has corrected and improved earlier versions; any remaining mistakes are of course my own. I wish to thank him for all this help and for informing me about improvements that he made in a recent unpublished manuscript. In the final version they have been partially replaced by still more recent unpublished results due to A. Ancona presented in Section 1.7 and at the end of Sections 3.2 and 4.1. I am grateful for his permission to include them here. I would also like to thank Anders Melin for his critical reading of a large part of the manuscript, and M. Andersson, M. Passare and R. Sigurdsson who agreed to the inclusion in Chapter IV of some material from an unpublished manuscript of theirs. Thanks are also due to the publishers and their referees. Lund in June 1994 Lars Hormander
CONTENTS
Preface
iii
Contents
v
C h a p t e r I. C o n v e x functions of one variable 1.1. Definitions and basic facts 1.2. Some basic inequalities 1.3. Conjugate convex functions (Legendre transforms) 1.4. The r function and a difference equation 1.5. Integral representation of convex functions 1.6. Semiconvex and quasiconvex functions 1.7. Convexity of the minimum of a one parameter family of functions C h a p t e r II. Convexity in a vector space 2.1. Definitions and basic facts 2.2. The Legendre transformation 2.3. Geometric inequalities 2.4. Smoothness of convex sets 2.5. Projective convexity 2.6. Convexity in Fourier analysis
1 1 9 16 20 23 26 28
finitedimensional 36 36 66 75 94 98 111
Chapter III. Subharmonic functions 3.1. Harmonic functions 3.2. Basic facts on subharmonic functions 3.3. Harmonic majorants and the Riesz representation formula 3.4. Exceptional sets
116 116 141
C h a p t e r IV. Plurisubharmonic functions 4.1. Basic facts 4.2. Existence theorems in L^ spaces with weights 4.3. Lelong numbers of plurisubharmonic functions 4.4. Closed positive currents 4.5. Exceptional sets
225 225 248 265 271 285
171 203
4.6. Other convexity conditions 4.7. Analytic functionals
290 300
Chapter V . Convexity w i t h respect t o a linear group 5.1. Smooth functions in the whole space 5.2. General G subharmonic functions
315 315 324
Chapter V I . Convexity w i t h respect t o differential operators 6.1. Pconvexity 6.2. An existence theorem in pseudoconvex domains 6.3. Analytic differential equations
328 328 332 344
C h a p t e r V I I . Convexity and condition (*) 7.1. Local analytic solvability for d/dzi 7.2. Generalities on projections and distance functions, and a theorem of Trepreau 7.3. The symplectic point of view 7.4. The microlocal transformation theory
353 353
Appendix. A. Polynomials and mult linear forms B. Commutator identities
391 391 396
Notes
403
References
407
I n d e x of notation
411
Index
413
372 375 382
CHAPTER I
CONVEX FUNCTIONS OF ONE VARIABLE S u m m a r y . Section 1.1 just recalls wellknown elementary facts which are essential for all t h e following chapters. Section 1.2 is devoted t o proofs of basic inequalities in analysis by convexity arguments. T h e Legendre transform (conjugate convex functions) is discussed in Section 1.3 in a spirit which prepares for t h e case of several variables in Chapter II. Section 1.4 is an interlude presenting an interesting characterization of t h e F function by t h e functional equation and logarithmic convexity, due to Bohr and MoUerup. We introduce representation of convex functions by means of Green's function in Section 1.5, as a preparation for t h e representation formulas for subharmonic functions. In Section 1.6 we discuss some weaker notions of convexity which occur in microlocal analysis. Section 1.4 and most of Section 1.6 can be bypassed with no loss of continuity. T h e last section. Section 1.7, studies when the minimum of a family of (convex) functions is convex. T h e extension t o (pluri) subharmonic functions in Chapters III and IV will be essential in C h a p t e r VII.
1.1. Definitions and basic facts. Let / be an interval on the real line R, which may be open or closed, finite or infinite at either end, and let / be a real valued function defined in / . Definition 1.1.1. / i s called convex if the graph lies below the chord between any two points, that is, for every compact interval J C / , with boundary dJ, and every linear function L we have (1.1.1)
sup(/L)=sup(/L). J
dJ
One calls / concave if — / is convex. Let dJ = {3:i,X2}. An arbitrary point in J can then be written Air^i A2a:2 where Xj > 0 and Ai + A2 = 1. Since L{XiXi + A2X2) = XiL{xi) A2i(x2), and we can choose L and a constant a with L \ a = f on 9 J , follows that (1.1.1) is equivalent to (1.1.1)' f{XiXi^X2X2) < Xif{xi)^X2f{x2), if Ai,A2 > 0, A1 + A2 = 1, Xi,X2 E
+ + it
/.
If / is both convex and concave, then there must be equality in (1.1.1)', that is, / = L + a where L is linear and a is a constant. Such a function
2
I. CONVEX FUNCTIONS OF ONE VARIABLE
is called affine] it can of course be uniquely extended to all of R. More generally, a map / between two vector spaces is called afEne if it is of the form f = L \ a with L linear and a constant. This is equivalent to (1.1.2)
/ ( A i x i + A2:r2)  Xif{xi)
+ A2/(x2),
when Ai + A2 = 1.
Indeed, ii L = f — /(O) we obtain L{Xx) = XL{x) when X2 = 0, hence L{xi + X2) = L{xi) + L{x2) follows if Ai = A2 =  . This means that L is linear. Conversely, if / = L + a with L linear we obtain not only (1.1.2) but more generally
(1.1.2)'
/ ( ^ X^x,) = 5 ^ X,f{xj),
if X ; A, = 1.
The following statements are immediate consequences of (1.1.1) or (1.1.1)': T h e o r e m 1.1.2. If fj are convex functions in I and Cj G R are > 0, J = 1 , . . . , n, then f — Y^ Cjfj is a convex function in I. T h e o r e m 1.1.3. Let fa, a E A, be a family of convex functions in I, and let J be the set of points x e I such that f{x) = sup^,^^/^(rc) is < lcxD. Then J is an interval (which may be empty) and f is a convex function in J. If fj, j = 1 , 2 , . . . , is a sequence of convex functions and J is the set of points x ^ I where F{x) — limj_,oo /j(^) < +00, then J is an interval and F is a convex function in J unless F = —00 in the interior of J or J consists of a single point. To prove the second statement one just has to write F{x) = lim.Fp^(x) where FN{X) = sup^^^y fj{x) and use the obvious first part. Exercise 1.1.1. Prove that one cannot replace sup by inf or lim by lim in Theorem 1.1.3. Exercise 1.1.2. Let / and J be two compact intervals with J C I and lengths /, I J, and let / be a convex function in / . Prove that if m and M are constants such that / < M in / and f > m in J then f>M{M
m)\I\/{\J\
+ d(J, dl))
in / ,
where d{J, dl) is the shortest distance from J to dl and the denominator is assumed 7^ 0.
DEFINITIONS AND BASIC FACTS
3
T h e o r e m 1.1.4. Let f be a realvalued function deGned in an interval I, and let ip be a function defined in another interval J with values in I. Then f o cp is convex for every convex f if and only ifcp is afEne; and f o cp is convex for every convex (p if and only if f is convex and increasing. Proof. li f ocp is convex for f{x) = x and for f{x) = —x^ then cp is both convex and concave, hence afRne. Conversely, if ^ is affine it is obvious that f o (p inherits convexity from / . Now assume that f o cp is convex for every convex cp. Taking (p{x) = x we conclude that / must be convex. If yi < y2 are points in / , then (p{x) = yi h (2/2 — 2/i)^ is convex in [—1,1] , and ii f ocp is convex it follows since fo(p{±l) = f{y2) and f o(p(0) — f{yi) that f{yi) < f{y2), so / must be increasing. Conversely, assume that / is increasing and convex, and let xi,X2 G / , Ai, A2 > 0, Ai + A2 == 1. Then f{(p{XiXi + A2X2)) < f{Xi(p{xi)
+ X2(p{x2)) < Xif{(p{xi))
+ A2/((^(x2)),
where the first inequality holds since cp is convex and / is increasing, the second since / is convex. This completes the proof. If xi < X < X2 then x = AiXi + A2X2 for Ai = (x2 — x)/{x2 X2 = {x — rci)/(j:2 — rci), so (1.1.1)' means that {x2  xi)f{x) (1.1.1)''
< {x2  x)f{xi)
{fix)  f{x,))/{x
\{x
xi)f{x2),
— xi)^
that is,
 X,) < {f{x2)  f{x))/{x2  X).
Hence we have: T h e o r e m 1.1.5. f is convex if and only if for every x E I the difference quotient {f(x •} h) — f{x))/h is an increasing function ofh when x \h E I and h ^ 0. Corollary 1.1.6. If f is convex then the left derivative f[{x) and the right derivative f!^{x) exist at every interior point off. They are increasing functions. If xi < X2 are in the interior of I we have (1.1.3)
//(:ri) < / ; ( x i ) < (/(X2)  fix,))/{x2
 x^) < //(X2) < flix^).
In particular, f is Lipschitz continuous in every compact interval in the interior of I.
contained
There is no need for / to be continuous at the end points of / , but f{x) has a finite limit when x converges to a finite end point of / belonging to / , again by Theorem 1.1.5. Changing the definition at the end points if necessary we can therefore assume that / is continuous also there. The right (left) derivative exists then at the left (right) end point but may be
4
I. CONVEX FUNCTIONS OF ONE VARIABLE
—oo (+00). If we allow / to take the value +00 we can always make the interval / closed. Using the continuity of / we obtain from (1.1.3) if xi < X2 are points in / (1.1.3)'
lim fl{xi
•\e)< U{X2)  f{x,))/{x2
 x,) < lim fl{x2  e).
If we let X2 i xi or xi t ^2? we obtain T h e o r e m 1.1.7. If f is convex in I and x is an interior point, (1.1.4)
/ ; ( a ; ) = lim / ; ( x + £ ) = limfUx
(1.1.5)
//(x)= l i m / ; ( x  £ ) =
We shall therefore write f'{x{0) conditions are equivalent
= f!,{x),
then
+ e),
\imflixe). f'{x  0) = //(a:). The following
(1) / / is continuous at x; (2) /^ is continuous at x; (3) frix) — fl{x), that is, f is differentiable at x. These conditions are fulfilled except at countably many Proof. The last statement follows from the fact that iixi in / , then E
points. < X2 are points
(/;(a;)//(^)) 0 one calls f strictly convex. E x a m p l e 1.1.11. f{x) = e^^ is a convex function on R for every a G R. If 7* > 1, then fr{x) = a:'^ is a convex function when x > 0, if r < 0 then fr is convex when x > 0, but if 0 < r < 1 then x'^ is concave when x > 0. The functions g{x) = xlogx and h{x) = — logx are convex when x > 0. Another immediate consequence of Theorem 1.1.9 and Corollary 1.1.6 is: Corollary 1.1.12. Convexity is a local property: If f is defined in an interval I and every point in I is contained in an open interval J C I such that the restriction of f to J is convex, then f is convex. We have defined convexity in terms of affine major ants, but there is also an equivalent definition in terms of affine minorants: T h e o r e m 1.1.13. A reaivaiued function f defined in an interval I is convex if and only if for every x in the interior of I there is an affine linear function g with g < f and g{x) = f{x). Proof. Assume that / is convex. Choose k E [fi{x), fri^)] and let g{y) = f{x) + k{y — x). Since g{x) — f{x) and (1.1.3) gives f{y) > fix) + iy
x)r,{x)
> g{x),
iiy>x;
f{y) > fix) + iy
x)f[ix)
> giy),
iiy<x,
DEFINITIONS AND BASIC FACTS
7
the necessity is proved. Now assume that / satisfies the condition in the theorem. We must prove that (1.1.1)' holds. In doing so we may assume that xi 7^ X2 and that A1A2 > 0, which impHes that x = AiXi f A2X2 is an interior point of / . If g is an affine minorant of / with f{x) = g{x) then 2
2
2
1
1
1
which completes the proof. In view of (1.1.2)' the second part of the proof gives a much more general result with no change other than extension of the summation from 1 to n: T h e o r e m 1.1.14. Let f be convex in the interval I, and let xi,... / . Then we have n
(i.i.ir
n
^x^ G
n
/(E^^^^)^E^^/(^^)' ifAi,...,A„>o, Y.^, = i. 1
1
1
If Aj > 0 for every j , then there is equality in (1.1.1)'" if and only if f is affine in the interval [minxj,maxxj]. E x e r c i s e 1.1.8. Prove (1.1.1)'" directly from (1.1.1)' by induction with respect to n. (1.1.1)'" is usually called Jensen^s inequality, and so is the following more general version involving integrals instead of sums: E x e r c i s e 1.1.9. Let / be a convex function in the interval / , let T be a compact space with a positive measure dji such that Jrpd/jL(t) = 1, and let x(t) be a // integrable function on T with values in I. Prove that
/ ( / X{t)dti{t)) < I f{x{t))dtL{t). JT
JT
E x e r c i s e 1.1.10. Let 11 be an orthogonal projection in a finite dimensional Euclidean vector space E. Show that if yl is a symmetric linear operator in E then
Tv(n/(nyin)n) < Tr(n/(^)n) for every convex function / (Berezin's inequality). (Recall that if B is a linear transformation in E then Tr B = Y^{Bej, Cj) if Cj is any orthonormal basis in E and (•, •) denotes the scalar product. If B is symmetric then f{B) has the same eigenvectors as B with every eigenvalue A replaced by /(A). — Hint: Express both sides in terms of the eigenvectors of IIAII in HE and of A in E.)
8
I. CONVEX FUNCTIONS OF ONE VARIABLE
E x e r c i s e 1.1.11. W^ith the notation in Exercise 1.1.10 show that for any / G C'{I) Mminf
< Tr(n/(A)n)  T r ( n / ( n A n ) n ) < Mmax/",
where M —  Tr ( n A ( I d  n ) A n ) and / is the interval bounded by the largest and smallest eigenvalues of A] Id is the identity operator. A number of applications of Jensen's inequality will be discussed in Section 1.2. We shall first end this section by discussing some seemingly weaker definitions of convexity which are sometimes useful. T h e o r e m 1.1.15. If f is continuous but not convex in the open interval I, then one can find ?/ G / , c G R and e > 0 such that (1.1.7)
f{y ih) < f{y) + ch  eh^,
when \h\ is small.
Proof. Let J = [a,b] C I be an interval such that for some affine g we have f < g on dJ but s u p j ( / — g) > Q. Then fs{x) = f{x)  g{x) + e{x  a){x  b) is < 0 in dJ but sup j /^ > 0 if e is small enough. The maximum is then taken at an interior point 7/ G J , so f{x)g{x){e{xa){xb)
= fe{x) < fs{y) = f{y)9{y)
+
e{ya){yb),
when X ^ J. With x = y \ h it follows from Taylor's formula that f{y ^h)
0 if / is convex.
SOME BASIC INEQUALITIES
9
Exercise 1.1.12. Let / be a continuous realvalued function in the interval / . Prove that / is convex if for arbitrary e^6 > 0 and x in the interior of / there is a positive^measure with support in [0,6] such that
/ {f{xhh)\f{xh)2f{x))dfi{h)>e
h'^dfi{h)^0.
Jo
Jo
We shall end the section with some more esoteric conditions for convexity. Exercise 1.1.13. Let / be a realvalued function in an open interval / such that / is bounded above on some open nonempty subinterval and (1.1.9)
/(i(a; + 2 / ) ) < i ( / ( x ) + /(y)),
x,y € I.
Prove that / is convex. (Hint: Prove in order the following statements: (1) ( L l . l ) ' is valid if Ai, A2 are rational numbers with a power of 2 as denominator. (2) / is bounded above on every compact subinterval of / . (3) / is bounded below on every such interval. (4) / is continuous. (5) (1.1.1)' is valid in general.) On the other hand there are unbounded functions satisfying (1.1.9) such as all functions / satisfying the functional equation f{x^y)
= f{x) + f{y),
x,2/GR.
It is well known that there are such functions which are not linear. However, they are not measurable, which is confirmed by the following: Exercise 1.1.14. Prove that if / is measurable and satisfies (1.1.9), then / is a convex function. (Hint: The set Ea = {x;x E I^fi^) < a} is measurable, and it has positive measure for some a. We have x E Ea ii X \ y E Ea and x — y e Ea ioi some y. Show that this implies that Ea contains an interval.) 1.2. S o m e basic inequalities. When combined with Jensen's inequality the convex functions listed in Example 1.1.11 yield some of the most important inequalities in analysis. T h e o r e m 1.2.1 (Inequality b e t w e e n geometric and arithmetic m e a n s ) . Ifaj > 0, A^ > 0, j = 1 , . . . , n, and ^ ^ A^ = 1, then n
(1.2.1)
n
n^'^Ev.'
10
I. CONVEX FUNCTIONS OF ONE VARIABLE
with strict inequality unless all aj are equal. Proof. With aj = e^^ the inequahty becomes n
n
exp ( 5 ^ XjXj) < Y^ Xj expXj, 1
1
so (1.2.1) follows from Jensen's inequality since x i^ expx is convex and not afBne in any interval. T h e o r e m 1.2.2. Ifaj
> 0, Xj > 0, j = 1,... ,n, and Xli A^ = 1, then
n
(1.2.2)
n
^ I
"' ^•^P>1'
EV.^(EM) 1
1
with equality only if all aj are equal. Proof. If we raise both sides to the power p this follows from the fact that X 1^ x^ is convex when x > 0 iip > 1, and is not affine in any interval. The righthand side of (1.2.2) is called the P mean of a = ( a i , . . . ,an) with weights A = ( A i , . . . , A^). More generally we define
(1.2.3)
A^p(a;A) = ( E A , < )
\
P ^ 0
1
When p ^ 0 we have, since Yl^ Aj = 1, n
n
pMog(;^A,a^)=pMog(^A,(l+ploga,+0(/))) 1
1 n
= X]Ajlogaj+0(^), 1
so M.p{a] A) becomes a continuous function of p for all p eR
if we define
n
(1.2.3)'
A^o(tt5 A) ~ TT^j^
{^^^ geometric mean).
1
M.i{a; A) is the arithmetic mean of a and A^_i(a; A) is called the harmonic mean of a, with weights A. When p > 0 we have maxttjA^ ^ < Mp{a]X)
< maxa^,
and when ;? < 0 we have mina^ < Mp{a;X)
< mina^A^
,
so we get Mp{a] A) ^ ^4^00(0; A) as p ^ ±00 if we define (1.2.3)''
A^_oo(o^;A) — minttj;
A^+oo(tt;A) = m a x a j .
Theorems 1.2.1 and 1.2.2 are now special cases of the following:
SOME BASIC INEQUALITIES
11
T h e o r e m 1.2.3. If aj > 0, Xj > 0, j = 1,... ,n, and J^l Xj = 1, then Mp{a] X) is a strictly increasing function ofp G [—oo, +CXD] unless all aj are equal; in that case Mp{a^ X) is this common value, for all p and X. Proof. Assume that all aj are not equal. By Theorem 1.2.2 applied with aj replaced by a j , g > 0, we conclude that Mq{a] X) < Mpq{a] A),
p > 1,
which proves the statement for p > 0. Since M.p{a;X)
=
{Mp{a';X))\
the statement follows for p < 0 also. If we drop the condition ^ ^ Xj — 1 and apply (1.2.2) with Aj replaced by Xjl Y^^ ^k: we obtain for arbitrary Aj > 0
1
1
1
Here p' is called the exponent conjugate to p; note that p + p' = pp'. This inequality gets a more familiar form if we replace Aj by &^ and aj by ajh^
^ , noting that p(\ — p') + p' == 0, which gives
T h e o r e m 1.2.4 (Holder's inequality). For arbitrary positive aj and bj we have ifp > 1, p^ > 1, 1/p + 1/p' = 1 n
(1.2.4)
n
, /
n
i / '
E«A(E«0 (E 0, bj > 0 for j = 1 , . . . , n . Then
(B«.+M')""s(E4)""+(E'^)"". ^>i. 1
with strict inequality early dependent.
1
1
unless a = ( a i , . . . , a^) and b = (&i, • • •, &n) ^^^ lin
Proof. The theorem is trivial if a = 0 or & = 0. Otherwise we can choose a > 0 and /? > 0 so that n
1/
n
1
1/
1
Then the convex function of A G [0,1] defined by
fiX) = Y.iaXaj + P{lX)bjr 1
is equal to 1 when A = 0 or A = 1, and it is not afline in [0,1] unless aa — /3b. Otherwise we conclude that /(A) < 1 when aA = /3(1 — A), hence A r p/(a + p) and aX = ap/{a + /?). Thus n
. ,
(5^(a,+6,r) ' < l / a + lM 1
which completes the proof.
SOME BASIC INEQUALITIES
13
Exercise 1.2.2. Derive Minkowski's inequality from Holder's inequality. We give two more exercises involving Jensen's inequality, which are related to the notion of entropy: Exercise 1.2.3. Prove that — X^^^jloga;^ < logn if 0 < Xj, j = 1 , . . . , n, and Y^i ^j — 1 (We define rrloga: = 0 when x = 0.) E x e r c i s e 1.2.4. Let 0 < Xjk^ j = 1 , . . . , J , k = l , . . . , i i r , and let E i = i Efc=i ^jk = 1. Prove that J
K
J
K
 X^ X! ^^^ ^^s ^ok 0, Xj > 0, j = 1,... ,n, and ^ ^ Xj = 1, then p I—> plog A^p(a; A) is convex on R and not affine in any interval unless all aj are equal. Note that the difference quotients at 0 are logjVtp(a; A), so Theorem 1.2.7 contains Theorem 1.2.3 in view of Theorem 1.1.5. For the proof of Theorem 1.2.7 we need a lemma to which we shall return several times in related contexts: L e m m a 1.2.8. If g is a positive function defined in an interval I, then logg is convex (resp. affine) in I if and only ift H^ e^^g{t) is convex (resp. constant) in I for every (resp. some) c G R. Proof. If log^ is convex, then t ^^ ct i log^(t) is convex. Since u y> e^ is increasing and convex it follows from Theorem 1.1.4 that t i> e^^g(t) is convex. Conversely assume that t y^ c^^g{t) is convex for every c. For every compact interval J C I the maximum of e^^g{t) when t G J is then assumed when t G 5 J , which means that the maximum of ct \ log g{t) in J is taken when t G dJ. Hence log^ is convex. The condition for log^ to be affine is trivial. Proof of Theorem 1.2.7. The convexity of the exponential function implies that p ^ e^^Mpia; Xy = ^
Xjie^ajf
14
I. CONVEX FUNCTIONS OF ONE VARIABLE
is a convex function, and it cannot be constant unless all aj are equal. Hence Theorem 1.2.7 is a consequence of Lemma 1.2.8. Corollary 1.2.9. Ifaj > 0, Xj > 0, j ^ 1,... ,n, and Y^'l Xj = 1, then X I—> log A^i/a;(a; A) is a convex function for x > 0 and a concave function for X < 0, and not affine in any subinterval unless all aj are equal. Proof. Let (p{p) = P^ogMp{f), which is a convex function by Theorem 1.2.7. The claim is that x H> tlj{x) = x(p{l/x) is convex for x > 0 and concave for x < 0, and this is clear since
^\x)
= ^{1/x)  ip\l/x)/x,
i^"{x) =
^"{\lx)lx'^.
Exercise 1.2.5. Prove that Corollary 1.2.9 follows from Holder's inequality Exercise 1.2.6. Prove that '0(/ o ip) is convex for every convex / if and only if 0 and ijjip are afRne and V^ > 0, unless ip is constant. (This explains the proof of Corollary 1.2.9.) Exercise 1.2.7. Prove that liui and U2 are positive functions such that log Til and log'U2 are convex, then \og(ui + U2) is convex. The mean values Mp{a\X) studied above can be generalized further. Let (/p be a strictly monotonic continuous function defined in an interval / , and let ip"^ be the inverse function defined in the range which is also an interval. If a i , . . . , a^ G / and A i , . . . , A^ > 0, J^^ Aj = 1, then ^ ^ Xj(p{aj) belongs to the range of (^, which makes
M c + k{y — x)
for every y G R.
Proof. This is a consequence of Theorem 1.1.13 and (1.3.1) if x is in the interior / of the interval where / < oo. If rr is, say, the right end point of this interval we choose ^ G / and note that since f{y)>f{0
+ {yOf'i^
+ o)
VyGR
the statement follows unless f{() h{x — 0 / ' ( ^ + 0) < c for all ^ G / . Since fiO "^ / ( ^ ) > ^ when ^ T ^5 tliis would imply that / ' ( ^ f 0) ^ —oo when ^ T X, which is absurd since / ' ( ^ + 0) is increasing. This proves the statement except when x ^ I. Then we just have to take k sufficiently large positive or negative. Definition 1.3.2. The Legendre transform (also called the conjugate function) / of / is defined by (1.3.2)
fiO = supix^ 
fix)).
X
The term Legendre transform may be preferable because of the ambiguity of the term "conjugate function". Note that only the interval where / is finite matters in the definition (1.3.2). By Theorem 1.1.3 it is clear that / is convex, for it is the supremum of a family of functions which are affine, hence convex; / is lower semicontinuous since the supremum of any family of lower semicontinuous functions inherits this property. Since X i> x^ — f{x) is increasing for x < XQ \i / ' ( X Q — 0) < ^, the restriction of / to [xo,oo) determines / on ( / ' ( X Q — 0),oo), if / < oo in a neighborhood of XQ. Thus / ( ^ ) is determined for large ^ if / ( x ) is known for large x. T h e o r e m 1.3.3. For every convex lower semicontinuous have the inversion formula f = fj that is, (1.3.3)
Proof
fix)
= sup(xe  / ( O ) .
Prom the definition (1.3.2) we obtain at once
Hence fix)
> sup(xe  fiO)
=
fix).
function f we
18
I. CONVEX FUNCTIONS OF ONE VARIABLE
So far we have not used the hypotheses on / . By Lemma 1.3.1 they imply that for every c < f{x) we have for somefcE R ky — f{y) < kx — c^
V?/ E R,
hence f{k) < kx — c.
Thus c < kx — f{k) < f{x)^ which proves that f{x) < f{x). The proof is complete. If / is differentiable we can determine / ( ^ ) by differential calculus in the interior of the interval where / < oo; this leads to elimination of x from the equations (1.3.4)
fix) + m=^i,
/'(^) = e
If / € C^ and / " > 0 the second equation (1.3.4) determines x locally as a C^ function of ^, and differentiation of the first equation (1.3.4) with respect to ^ then gives / ' ( O = oo, so f e C^ (locally) and we obtain the symmetric formulas (1.3.5)
fix)+fiO=^^^,
f'{x)=C,
f'iO=x
Exercise 1.3.1. Find the Legendre transform of the following functions: a) fix) = \x\^, p > 1; b) fix) = e^; c) fix) = x l o g x ii x > 0, /(O) = 0, fix) = +00 if X < 0. Conjugate functions are often presented in a different way in the literature. Let / be a strictly increasing continuous function on [0, CXD) with /(O) = 0, and denote the inverse function by g. Set
J^fit)dt,
ifrr;>0;
^^^ ~ \ J^ gis) ds,
if ^ > 0.
Then F and G are conjugate convex functions. In fact, if J: > 0, ^ > 0 then x^ — Fix) is the area of the rectangle with vertices (0,0), (0, x), (a;, ^), (0,^) minus the area of {(/;,?/); 0 < y < fit),0 < t < x}. For given ( it is clear from a picture that it grows until fix) = (, and decreases afterwards, so the maximum value is equal to Gi^). Hence F(^) = G(^) when ^ > 0, and Fi^) — 0 when ^ < 0, with the maximum taken for x = 0. E x a m p l e 1.3.4. With fix) = x ^  \ p > 1, we obtain ^ ( ^ = C^'~^ where 1/p f 1/p' = 1, for this is equivalent to (p — l ) ( y — 1) = 1. Hence Fix) = x^/p for X > 0 and G(<J) = (^'^ /p' for (^ > 0 are conjugate functions. Similarly Fix)  e^  1  x for x > 0 and G ( 0 = ( 1 + 0 log(l + 0  ^ for ^ > 0 are conjugate functions. (Compare with Exercise 1.3.1.)
CONJUGATE CONVEX FUNCTIONS
19
If / and / are conjugate functions, we have for arbitrary a^, bj^ j = l,...,n, n
(1.3.6)
n
^ajb, I
n
< ^ / ( a , ) + J2f{b,). l
l
The first case in Example 1.3.4 gives, if aj > 0, bj > 0
1
1
1
Hence n
Y^ajbj
< 1,
if A = l,
B^l,
1
which imphes Holder's inequality. The inequality (1.3.6) becomes a substitute when other means are known. Some minimum problems can be solved by means of Legendre transforms. We give an example: P r o p o s i t i o n 1.3.5. Let f be an everywhere Unite convex function on R such that f{x) ^ Hoo as \x\ ^ oo. Let 5 i , . . . , 5 ^ be given positive numbers, let ti^... ^tn be given numbers ^ 0, and set n
(1.3.7)
n
Fs,t{A) = inf { 5 ^ s J ( a O ; a, G R, J ^ t.Siai = A}. 1
1
Then Fs^t is a continuous convex function with Legendre
transform
n
FsA^) = Y^Sif{ati). 1
Proof. Since / ^ Hoo at cx) it is clear that the infimum in (1.3.7) is attained in a fixed compact set when a bound for A is given. Hence the continuity follows. If ^^ tiSidi ~ A and J^^ tiSibi = B, then n
Y^tiSiiXa^
+ (1  X)bi) = A^ + (1  A)S,
1 n
n
n
Y^ SifiXa, + (1  X)bi) <XJ2 Si/(«^) + (1  A) ^ 1
1
Sifibi),
1
which proves the convexity. By definition we have n
= snp{aA  Fs^t{A)) = sup ^ ^ 1
which proves the proposition.
n
atiSiai  ^ 1
n
sj(ai)j = ^
Sif{ati), 1
20
I. CONVEX FUNCTIONS OF ONE VARIABLE
1.4. T h e r function and a difference equation. is defined by the Eulerian integral
The F function
/*CX)
(1.4.1)
T{x) = /
eH''^ dt,
x>0.
The convergence is obvious, and integration by parts yields the functional equation
(1.4.2)
r{x + i)=^xr{x),
x>o.
Since (1.4.3)
r ( i ) = 1,
we have T{n) ~ {n — 1)\ for every positive integer, so the F function interpolates the factorial to noninteger arguments. There are many functions satisfying (1.4.2) and (1.4.3), for these properties are preserved if we multiply T{x) with any function of period 1 which is equal to 1 when x = 1. However, one can characterize the F function uniquely by a convexity property: T h e o r e m 1.4.1. log F is a convex function on the positive real axis, and there is no other positive solution of (1.4.2), (1.4.3) having this property. Proof. To prove that logF is convex it suffices by Lemma 1.2.8 to prove that X — f > e^^F(x) is convex for every c G R. This follows at once since J2
poo
—^(e"^F(x))= / rfx^ Jo
e  ' ( c + logt)2e^^t^irf^>0.
(Instead of differentiating we could interpret the integral defining e^^F(a;) as a limit of sums of exponential functions and use Theorem 1.1.2, which is also true for "continuous sums".) To prove the uniqueness we first note that (1.4.2) can be written (1.4.2)'
logF(x  h i )  logF(x) = logrc,
where the righthand side is a concave function which is o{x) as x ^ cx). The uniqueness is therefore a special case of the following: T h e o r e m 1.4.2. Let h{x) be a concave function on {x E Ii]x such that h{x)/x —> 0 as x ^ oo. Then the difference equation (1.4.4)
g{x + l)g{x)
= h{x),
x > 0,
> 0}
THE r FUNCTION AND A DIFFERENCE EQUATION
21
has one and only one convex solution g with g{l) = 0, and it is given by nl
(1.4.5)
g{x) = h{x)
+ lim (xh{n) + Y^ihij)

h(x\j))).
Proof. The function h is increasing, for {h{x^y) — h{x))/y and ^ 0 as 7/ ^ oo for fixed x. Moreover, if ^ > 0 then (1.4.6)
{h{x + 2/)  Hx))/y
< {h{x)  h{x/2))/{x/2)
^ 0
is decreasing
as x ^ cx).
In particular, the sequence a^ — /i(n + 1 ) — h{n) is decreasing and converges to 0 as n ^ oo. Now (1.4.4) and the condition ^(1) = 0 give nl
(1.4.7) g{x)\h{x)xh{n)^{h(j)h{x\j))
=
g{x\n)g{n)xh{n).
If g is convex and k is an integer with x < k, then h{n  1) == g{n)  g{n  1) < {g{x + n) < {g{k + n)  g{n))/k
g{n))/x
= {h{n) \• • •jh(n +k 
l))/k.
Thus xani
< g{x + n)  g{n)  xh{n) < x{{k  l)a^ H
f 0^+^2)/^,
which proves that g{x hn) — g{n) — xh{n) —> 0 as n ^ 00. Hence it follows from (1.4.7) that g must be of the form (1.4.5). This uniqueness suffices to complete the proof of Theorem 1.4.1, but to prove Theorem 1.4.2 we must also show that (1.4.5) converges to a convex function with ^(1) — 0 satisfying (1.4.4). Only the convergence needs some motivation, for it is clear that ^(1) = 0, and (1.4.4) will follow since h{n) — h{x + n) ^ 0 as n —> 00, by (1.4.6). We write the limit (1.4.5) as Yl^ '^n(^)> where uo{x) = xh{l) — h{x) and y^n{x) — x{h{n + 1) — h{n)) { h{n) — h{x + n),
n > 0.
It is clear that Un is convex. Using the concavity of h we also obtain when k >X h(n — 1) — h{n) < {h{n) — h{x + n))/x x{an
 an~l)
< Un{x)
< X (ttn ~ {an {
< {h{n) — h{k + n))/k^ h an+fcl)/^),
n > 0.
hence
22
I. CONVEX FUNCTIONS OF ONE VARIABLE
Thus Un > 0, and since oo ^
k (ttn 
( t t n H • • • +
flnffcl)/^)
= ^
1
% ( 1 " J/k)
< OC,
1
the convergence of ^ ^ Un{x) follows, and the sum is convex since the terms are. This completes the proof. If we apply (1.4.5) to the F function, we obtain the product formula (1.4.8) ^ ^
r ( x ) = lim — ^, , ^ ^ n>oox(xf l )    ( x + n)
Using the concavity of h we can also get estimates for g. First note that for every x > 0 px + l
^{h{x) + h{x + 1)) < /
h{t) dt < h{x + I ) ,
for h has an af&ne majorant equal to /i at :r +  and an affine minorant equal to ft at a; and x + 1. Hence nl
xh{n)^Y.{h{j)h{x^j)) 1 nn — l
< xh{n) + /
px\n—\
h(t) dt f  ( / i ( l ) f /i(n  1))  /
/•cc+l
/i(t) dt
px\n—^
= / ft(^) rfi^ +  / i ( l ) +  / i ( n  1)  /
/i(t) 6/^ + xft(n).
It follows from (1.4.6) that the sum of the last three terms —> 0 as n —> oo. Similarly, nl
xh{n) +
Y.{h{j)h{x^j)) 1 rn—\
> xh{n) + / J\
= /
nx+n—l
h{t) dt
h{t) dt  \(h{x f1) h /i(x + n  1)) Jx\l
h{t) dt  \h{x + 1) 
/
h{t) dt + xh{n)  \h{x h n  1).
Again the sum of the last three terms ^ 0 as n ^ oo, so we obtain /
h{t) dt  ^h{x + 1) < g{x) + h{x) < /
h{t) dt +  / i ( l ) .
INTEGRAL REPRESENTATION OF CONVEX FUNCTIONS
23
We estimate the lefthand side from below by noting that \h{x^l)
/
h{t)dt=
/
{h{x^l)h{x\ t))dt
is decreasing, so we have proved the following estimate: T h e o r e m 1.4.3. For the convex solution of the difference (1.4.4) with g{l) = 0 given by Theorem 1.4.2 we have
(1.4.9) \g{x) + h{x) I
h(t)dt\ < \h{l)
equation
I h(t)dt < (/i(l)/i()).
Applied to the F function this estimate gives 24 < r ( x ) x e ^ ( x +  ) " ' ^ ~ 2 < 2 4 . Stirling's formula gives a much more precise result for large x: __
r{x)e''x
. 1
>
—
^"^2 > V27r,
as rr ^ oo.
It can be further improved by Stirling's series. However, this has little to do with the topic of convexity. Exercise 1.4.1. Prove under the hypotheses of Theorem 1.4.2 the existence of the limit rx\
lim {g{x) ^h{x) 
a:;—>+oo
I
h /i
2
h{t)dt).
•^2
1.5. Integral representation of convex functions. Let I = (a, 6) be a bounded open interval and let / be a convex function which is bounded in / . We can define / ( a ) and f{b) so that / is continuous in / = [a, b]. Set ., ^ , . (1.5.1)
^ . . ( ix~b){ya)/{ba), when a < y < x < b, Gi{x,y) = < / / /A t (x — a){y — b)/[b — a), when a < x < y < b.
Note that Gj is continuous and < 0 in the square I x I^ and that Gj = 0 on the boundary. Furthermore, Gi{x,y) is symmetric in x and y and affine in each variable outside the diagonal where the derivative has a jump equal to 1. In particular, Gj{x,y) is therefore a convex function in each variable when the other is fixed.
24
I. CONVEX FUNCTIONS OF ONE VARIABLE
T h e o r e m 1.5.1. There is a uniquely determined positive measure dfi in I such that (1.5.2)
fix) 
/ Gjix,y)dfx{y)
+ ^f{b)
JJ
+ ^ / ( a ) ,
0— a
x
el.
a— 0
In particular, (1.5.1) implies that (1.5.3)
{x a){b  x)d[i{x) < oo.
Conversely, for every positive measure satisfying (1.5.3) the integral in (1.5.2) defines a continuous convex function f in I which vanishes on dl, and f" = dpi in the sense of distribution theory. Proof. Assume at first that / is convex in a neighborhood of / . Then ^(x) = fr{oo) is an increasing function in / . Since Gj = 0 on the boundary, an integration by parts gives / Gi{x,y)dfi{y)
= 
dGi{x,y)/dyfi{y)dy
= F~; nr{y)dy'^ ba
J^
fK{y)dy
ba J^ 0—a
0— a
(cf. Theorem 1.1.9) which proves (1.5.2). If we choose x = {a\ b)/2 and let d{y) — {b — a) 12 — \y — {a{ &)/2 be the distance to the complement of / , we obtain from (1.5.2) (1.5.4)
^ d{y)d^l{y) = / ( a ) + /(&)  2 / ( i ( a + b)).
Dropping the assumption that / is convex in a neighborhood of / we can apply (1.5.4) to the interval {a\ £,b — e) for small e > 0, and when e > 0 we conclude that (1.5.4) holds for / , which implies (1.5.3). When e ^^ 0 we also conclude that (1.5.2) is valid. li (f e CQ{I) we have found that ip{x) = /
Gi{x,y)ip"{y)dy
without any convexity assumption on if. If / is defined by (1.5.2) with d/jL >0 satisfying (1.5.3), it follows from the convexity of x \^ Gi{x^y) that / is convex, and / f{x)(p"{x)dx Ji
=
Gi{x,y)(p'\x)dxdfx{y) JJixi
= / ip{y)dii{y). Ji
INTEGRAL REPRESENTATION OF CONVEX FUNCTIONS
25
Here we have used the symmetry of Gj. Thus d/x = f" in the sense of the theory of distributions which proves the uniqueness. Until now we have only used differentiation in a classical sense, and this would still suffice here. Thus we have not underlined the fact that for a convex function / in an open interval / , the derivatives /^ and / / both define f in the sense of the theory of distributions. This follows at once since for a nonnegative test function if G CQ{I) we have by monotone convergence, for example, [ f^{x)ip{x)
dx = lim
l{f{x\h)f{x))h^ip{x)dx
= lim / f{x){(p{x — h) — ip(x))h~^ dx = —
f{x)(p\x)
dx.
In analogous discussions of subharmonic functions later on, the language of distribution theory will be much more essential. Theorem 1.5.1 means that every convex function in a finite interval is a superposition of a linear function and functions of the form x \^ G{x^y)^ or equivalently, x \^ (x — y)^ oi x y^ \x — y\^ where t__ = max(t, 0) when t G R. Jensen's inequality N
N
N
is trivial i f / ( x ) = 1 or f{x) ~ ±x^ and it follows from the triangle inequality when f{x) = \x — y\. Hence it is true in general. Similarly, Berezin's inequality (Exercise 1.1.10) follows if we prove that
Tr(nnAnH) < Tr(n^n). li Si,... jSi^ G HE is an orthonormal basis of H ^ consisting of eigenvectors for HAH, then
1V(HHAHH) = 5 ] (  H A H  . „ . , ) = 1
Y^\{UAUe,,ej)\ 1
= Y, \iAei,e,)\ 1
< Y,{\A\e^,e^)
= Tr {U\A\U).
1
We give as an exercise to prove another inequality due to Berezin:
26
I. CONVEX FUNCTIONS OF ONE VARIABLE
Exercise 1.5.1. Prove that if £" is a finitedimensional Euclidean vector space and A i , . . . , AN are positive symmetric maps in E with ^_ Aj = Id, the identity, then N
TV f{Y,AjXj) 1
N
< TV 5 ] A,/(x,),
xj eJ, j =
h...,N,
1
provided that / is convex in the interval J C R. (Hint: Prove first that if A is symmetric then TV \A\ = max5 f in dJ, then L{t) > f{t) when a > t E J, since / — L is decreasing. Hence / ( a ) = / ( a h 0) < L{a) by condition (iii), if a G J , and the convexity gives that f < L when a < t £ J. If / is monotonic the necessity is also obvious. To prove the necessity otherwise we note that if ti < ^2 and f{ti) > 7(^2), then f{t) > / ( t i ) for t < ti since f{t) < f{ti) > /(^2) would contradict the definition even with L = 0. If we denote by a the supremum of all ti e I with f{ti) > f{t2) for some ^2 > ^i, ^2 ^ It it follows that / is increasing to the right of a and decreasing to the left of a. The lower semicontinuity gives /(a)<min(/(a + 0),/(a0)). If / ( a ) < f{a\ 0) we get a contradiction, for if a < t G / and f(t) > f{a), then / ( ( I  e)a + et) < (1  e)f{a) + ef{t) > / ( a )
SEMICONVEX AND QUASICONVEX FUNCTIONS
27
when e ^ +0. This completes the proof. Semiconvexity is of course invariant under composition with increasing affine functions. It does occur naturally in some analytic contexts; see ALPDO [IV, pp. 145147]. In the beginning of the proof we actually encountered an even weaker concept, which does not depend at all on the affine structure of R but only on the notion of intermediate point, so that it is invariant under composition with monotonic functions. Definition 1.6.3. A function / defined on an interval / C R with values in R will be called quasiconvex if / is lower semicontinuous and for every compact interval J C I the equality (1.1.1) holds when L is a constant, say 0. There is a description analogous to Theorem 1.6.2: T h e o r e m 1.6.4. / is quasiconvex in the interval / C R if and only if either (i) / is decreasing and continuous to the right, or (ii) / is increasing and continuous to the left, or (iii) there is a point a e I such that f satisfies (i) in I fl {—oo,a) and (ii) in I n (a, +oc), and f{a) < m i n ( / ( a + 0), / ( a  0)). Proof. The sufficiency is obvious and the proof of necessity was a part of the proof of Theorem 1.6.2. In spite of the fact that the notion of quasiconvex function seems quite trivial, it has a prominent role in the theory of linear partial differential operators, although in a somewhat different guise. In fact, if/ is differentiable at every point then / is quasiconcave if and only if (1.6.1)
/ ' ( ^ ) l^^s no sign change from h to — for increasing x.
This is closely related to the socalled condition ( ^ ) . (See Definition 7.3.3.) Remark. In the applications referred to, the natural continuity condition is upper semicontinuity and not lower semicontinuity. It has been changed here to agree with the standard condition for convex functions. However, upper semicontinuity has some obvious advantages such as fixing / ( a ) as m a x ( / ( a h 0), / ( a  0)) in Theorem 1.6.4. T h e o r e m 1.6.5. Semiconvexity is a local property: If f is defined in an open interval / C R and for every point x e I there is an open interval J with x e J C I where f is semiconvex, then this is true in I. Proof. Let J be a maximal open subinterval of / where / is strictly increasing and convex. If J is not empty then the right end point XQ is
28
I. CONVEX FUNCTIONS OF ONE VARIABLE
equal to that of / . In fact, if XQ E / then / is by hypothesis semiconvex in a neighborhood of XQ, and Theorem 1.6.2 proves that the convexity extends to a larger interval. Hence there can only be one such maximal interval J , and / must be decreasing to the left of it, again by Theorem 1.6.2. Quasiconvexity is not a local property, for a locally quasiconvex function can be monotonic in a number of intervals separated by intervals where it is constant. However, if / is locally quasiconvex and not constant in any open interval, then / is quasiconvex. We leave the verification to the reader. 1.7. Convexity of t h e m i n i m u m of a one parameter family of functions. The maximum of a family of convex functions is convex by Theorem 1.1.3, but the minimum is usually not. However, the following theorem shows that convexity of the minimum of a family of functions, convex or not, is decided by local conditions. T h e o r e m 1.7.1. If X C R is an open interval, / = [a, 6] C R is a compact interval, and u ^ C'^{X x I), then U{x) = mint^/ u{x,t) is convex in X if and only if the following three conditions are fulfilled: (i) Ifx G X I;u{x,t) (ii) Ifx eX (iii) Ifx GX (1.7.1)
then u'^{x,t) does not depend on t when t G J{x) = {t E = U{x)}. andtG J{x) then u'^^{x,t) > 0. andte J{x) \ dl then < , ( x , t) + 2 < , ( x , t)X + < ( x , t)X^ > 0,
A G R.
Then U G C^'^(X), that is, U G C^(X) and U' is locally Lipschitz uous. IfY is the open subset of X defined by (1.7.2)
Y = {x e X;t ^ dl and u'^ti^,t) > 0 when t G J{x)},
then U G C^iY) (1.7.3)
contin
and
U"{x) = min K J x , t )  < , ( x , t ) V < ( ^ , ^ ) ) .
For almost all x e X (1.7.4) U"{x) = u'^,{x,t)  u'^,{x,tflu'l,{x,t)
^^Y
ift G J{x) \ dl and < ( x , t ) > 0,
U"{x) = u'^^{x,t) ift G J{x) and t G dl or < ( x , t ) = 0. Proof. First we prove that the conditions (i)(iii) are necessary. If U is convex then for every x G X there is some c G R such that U{x \ h) > U{x) + ch when x\h e X, hence u{x \h,t~[ s) > u{x, t) + ch,
if t G J{x),
x + h E X, th s e I.
CONVEXITY OF THE MINIMUM OF A FAMILY
29
With 5 = 0 this imphes u'^{x,t) — c and u'^^{x^t) > 0 since X is open, which proves (i) and (ii) (and also the uniqueness of c, so that U ^ C^). li t ^ dl we also obtain u'f.{x^t) = 0 and that the second differential of u is nonnegative at (x,t), which proves (iii). (The condition (1.7.1) is also necessary at points t G J{x) fl dl where u'f.{x^t) = 0.) To prepare the proof of sufficiency we prove a lemma which also clarifies the role of the condition (i) in Theorem 1.7.1: L e m m a 1.7.2. Let X cH be an open interval and I a compact set. If u E C{X X / ) then U(x) = min^^/ u{x, t) is continuous. If X 3 x H^ U{X^ t) is in C^ fortel and u'^ is continuous in X x I^ then U is locally Lipschitz continuous in X. We have U G C^{X) if and only ifu'^{x, t) is independent oft G J{x) = {t E I;u{x,t) = U{x)}, and then we have U'{x) = u'^{x^t)
when t G J{x)^ x G X.
If in addition I is finite and x H^ u{x^t) is in C^ then U G C^(X) and
U'\x)=
min < , ( x , t ) ,
xeX.
If I = / i U /2 where Ij are disjoint, and X 3 x \^ u{x, s) — u{x, t) is convex when 5 G / i and t G /2, then either U{x) = miute/i u{x, t) for all x E X or U{x) — mintG/2 u{x,t) for all x ^ X. Proof. Ii K C I is a compact subinterval then the uniform continuity of u in K X I implies that U is continuous, and ii \u'^\ < M in K x I then M is a Lipschitz constant for U in K. If C/ G C^ then the derivative of the nonnegative function u{x,t) — U{x) with respect to x must vanish at every zero, that is, when t G J{x), so u'^{x^t) — U'{x) for every t G J{x). Conversely, if u'^{x,t) is continuous and independent of t when t G J{x) and V{x) is defined as this common value, then V is continuous since {{x,t) e X X I;t e J{x)} is closed. By Taylor's formula U{x + h) < min u{x \h,t) < U{x) + V{x)h f o{h). t£J{x)
For every ^ > 0 we can find a neighborhood u of J{x) in / where \u'^{x, t) — V{x)\ < 6. For sufficiently small h we obtain U{x \ h) = m.inu{x 4 ft, t) > U{x) 4 m.inu'^{x, t)h f o{h) >U{x)
+
Vix)h6\h\+o{h),
which proves that U is difFerentiable at x with derivative V{x).
30
I. CONVEX FUNCTIONS OF ONE VARIABLE
Now assume that / is finite and that u{x, t) is a C^ function of x. To prove the last statement, assume that the set F = {x G X; U[x) = minu{x,
t)}
is not empty. It is obvious that F is a closed subset of X . li x E F and U{x) — u{x^s) for some 5 G / i , we choose t £ I2 such that u{x^t) = U{x) = u ( x , s ) , hence u'^{x^t) — u'^{x,s). Then the convex function X 3 X 1^ u{x, s) — u{x, t) is nonnegative in X , so s can be dropped from / in the definition of U. But this implies that x is an interior point of F , so F is open and closed, hence equal to X. Let xo G X and let /Q be the set of all t G J{XQ) such that u'^^{x{)^t) has the minimum value a. From the result just proved it follows that there is a neighborhood XQ of XQ in X such that U{x) = mint^/o ^(^5^)5 hence IQ n J{x) 7^ 0, when x G XQ. Since 7i^(x, f) — u\xo, t) — a{x — XQ) = o{x — xo),
t G /Q,
it follows that U'{x) — U'{XQ) — a{x — XQ) = o(x — XQ), hence U"{XQ) — a. Thus lJ"{x) = u'^^{x^t) for some t £ IQ when x G XQ, so U" is continuous at Xo, which completes the proof. End of proof of Theorem 1.7.1. From condition (i) and Lemma 1.7.2 it follows that U G C^{X) and that U'{x) = u'^{x,t) for every t G J ( x ) . If Xo G y then J(xo) is a finite set { t j , . . . , t ^ } . By the implicit function theorem the equation 'uj(x, t) = 0 has for A: = 1 , . . . , A/' a unique C^ solution t — tk{x) in a neighborhood of XQ such that t^(xo) = t^, and J ( x ) = { t i ( x ) , . . . ,tiv(3:)} then. Thus Y is open, and in some neighborhood of Xo we have U{x) = in.miemY, (b) {x E E'^ J{x) n dl{x) 7^ 0} is discrete, hence countable. In fact, if x^y E E and a G J{x) fl J{y) then
(C/'(2/)C/'(x))+ 0 when t = v?(x), X £ LJ. Then U{x) < u{x, ^{x)) = v{x), x e uj, and v G C^(a;). In F = {x e u;]v{x) = U{x)} we have v'{x) = C/'(x), hence v"{x) — U"{x) iix is not an isolated point in F and U' is differentiable at x. Isolated points in F are countable, and we can account for all (x, t) £ Xx (a, 6) with u^{x^ t) = 0, u'l^{x^t) > 0 by countably many functions ip as just considered. Since v'[x) = u'^{x,t) and v"{x) = u'^^{x,t)  \y
x\/N,
if \y  x\ < 1/N.
If y is also in E^ then \U'{y)  U'ix)  u'Ux, t{x))iy  x)\ < o{\x y\
+ \tix) 
t{y)\)
by Taylor's formula, for U'{y) = Uy{y^t{y)), U'{x) = n^(x,t(x)), and IAJ'^(X, t{x)) = u'^ti^^ ti^)) — 0 by definition oiE^ and condition (iii). Hence we can for fixed N choose (^ > 0 so that for x^y E EN \y  x\/N
< il/2N){\x
~y\ + \t{x)  t{y)\),
if \xy\
0 so that x i^ U'{x) + Ax is increasing in a neighborhood of K^ which means that U{x) + \Ax'^ is
34
I. CONVEX FUNCTIONS OF ONE VARIABLE
convex there. If we apply Theorem 1.7.1 to v{x^t) = u{x,t) f ^Ax"^ then (i), (ii) follow from (i), (iii) there, for (1.7.1) means that v^^ — u'l^ > 0 and that < ? = Q in X, and that ul^l^
 u'^^^ > 0
inX
when u[ = 0.
Let I C H be a compact interval and set ujix) — inf u(x, t), when I HTx ^ ^. teinT^ If the interval Jx = {t E I CiTx] u{x^ t) = ui{x)} is compact and nonempty when I nTx 7^ 0, then uj is a convex function in every component of the open subset of R where it is defined. Proof. Let XQ be a point such that iHTx^ ^ 0. We can choose a compact interval K such that
Jxo CK CinTx,,
dKnJx,
Cdl,
CONVEXITY OF THE MINIMUM OF A FAMILY
35
for TxQ is open. Then we have u{xo^ t) > uj{xo)^
when t G dK \ 9 / ,
and we can choose an open neighborhood u of XQ such that u x K C X and u{x, i) > u{x, s), ii X e (jj, t e dK \ (9/, s E JXQBy the quasiconvexity of u it follows that UT(X) = inf u(x,t), ^ teK ^ ' ''
X E u,
hence uj is convex in u by Corollary 1.7.4. Remark. If/ is any interval such that the hypotheses in Corollary 1.7.5 are fulfilled with / replaced by any compact subinterval, then it follows that ui is convex in the intervals where it is defined. In fact, if Ij are compact subintervals increasing to / , then uj. j uj^ and since uj. are convex it follows that uj is convex.
CHAPTER II CONVEXITY IN
A
FINITEDIMENSIONAL VECTOR
SPACE
Summary. Section 2.1 presents basic facts on convex sets — such as convex hulls and extreme points, intersection and separation properties — and on convex functions. It ends with some convexity properties of hyperbolic polynomials which will be important in Section 2.3. The Legendre transformation is extended to several variables in Section 2.2, where we also give some applications to game theory and linear programming. The role of the Legendre transformation in Fourier analysis is discussed in Section 2.6. The main topic in Section 2.3 is inequalities between mixed volumes, in particular the BrunnMinkowski and FenchelAlexandrov inequalities. A related result of H. Weyl on the volume of tube domains in a Euclidean space is also given. Section 2.4 is a brief discussion of the smoothness properties of projections of convex sets, and in Section 2.5 we study convexity in a projective rather than an affine space.
2 . 1 . D e f i n i t i o n s a n d basic facts. In Chapter I convex functions were always assumed to be defined in intervals. When we pass to functions of several variables we must first introduce convex sets which will replace the intervals. Definition 2.1.1. Let F be a vector space over R. A subset X of F is called convex if every line intersects X in an interval, that is, (2.1.1) Ai^i + A2a;2 G X
when Ai, A2 > 0, Ai + A2 = 1, and a:i,a;2 G X.
A function / : X —> (  0 0 , +00] is then called convex if (2.1.2)
/ ( A i x i + A2X2) < A i / ( x i ) + A2/(x2) when Ai, A2 > 0, Ai 4 A2 = 1, and 0:1,3:2 G X.
The condition (2.1.2) means precisely that the epigraph of / , {(x, t) eVeR;x
eX,t>
f{x)},
is convex. We can therefore concentrate our discussion on convex sets and obtain results on convex functions as corollaries.
DEFINITIONS AND BASIC FACTS
37
Exercise 2.1.1. A subset X of F is called starshaped with respect to XQ if XXQ + (1 — X)x E X for every x E X and 0 < A < 1. Show that for every X the set of XQ such that X is starshaped with respect to XQ forms a convex set (which may be empty). The condition (2.1.1) implies a seemingly stronger one, N
(2.1.1)'
N
YJ ^3^3 ^ ^
if >^3 > 0^ XI \ = 1^ ^3 ^ ^•
1
1
This follows by induction with respect to N. For let iV > 2, Ajv 7^ 1 and assume that (2.1.1)' has been proved with N replaced by iV — 1. Set fjij = Xj/{1 — Aiv)j 3 < N. Then ^^ ~ IJLJ = I so J^i " /^j^j G X by the inductive hypothesis, hence Nl
X = {1 — Ajv) 2Z ^3^3 "^ ^N^N G X.
It follows at once that we can make a corresponding extension of (2.1.2). In Section 1.1 we touched briefly on notions of affine geometry. It will be useful to make some additional remarks here before proceeding with the main topic. A subset W of the vector space V is called an affine subspace if Wo = {xxo;x
G W}
is a linear subspace of V for some XQ eV. This requires that XQ eW^ if xi is any other element of W it follows that Wo = {x xi,x
and
G W},
for a; — a: 1 = x — xo — {xi — XQ). The definition as well as Wo is therefore independent of the choice oi xo E W. In view of the definition of a linear subspace of V it follows that W is an afiine subspace if and only if for arbitrary Ai, A2 G R and xi^X2 £W we have Ai(xi  xo) + A2(x2  xo) + xo G VF, which implies that (2.1.3)
Aio^i 4 A2a;2 eW
if Ai + A2  1, x i , 0:2 G W.
38
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
The proof that (2.1.1) imphes (2.1.1)' also shows that (2.1.3) is equivalent to the seemingly stronger condition N
(2.1.3)'
N
Yl ^J^J ^ ^
if Z l ^i " ^' ^^ ^ ^ •
1
1
When N = 3 this means that Ai^i + A2X2 + (1  Ai  A2)x3 = Ai(xi  xs) f A2(a;2  X3) \xs
eW,
which shows that (2.1.3)', hence (2.1.3), is sufficient for W to be an affine subspace. The dimension of W is by definition equal to the dimension of WQ. If xo,xi,... ,Xn G W, then xi — XQ, ... ^Xn — XQ are linearly independent if and only if Y^ Xj{xj  xo) = 0 =^
Xj = 0, j =
1,..., n.
This means precisely that XQ, ... ,Xn G W are affinely independent in the sense that n
n
The dimension of VF is thus one less than the supremum of the number of affinely independent elements. If XQ, •.. ,Xn G W are affinely independent and n = dimVF, then every element x e W can be written uniquely in the form X
= J2^^^^
where ^ A ^  1 ,
0
for this is equivalent to x — XQ = Y^i ^ji^j ~ ^0) and AQ = 1 — ^ ^ Xj. We shall therefore call XQ,. .. ,Xn an affine basis. If Vi and V2 are vector spaces then a map / : Vi —> V2 is called affine if the graph {(x, / ( x ) ) ; a: G Fi} is an affine subspace of Vi 0 V2, that is, n
(2.1.4)
n
/ ( X ; A,^,) = 5 ] A,/(a;,), 1
1
n
if x, G Fi, j ; A, = 1. 1
For / to be affine it suffices that this condition is fulfilled with n = 2. (See also Section 1.1.) Note that (2.1.4) also defines the notion of affine map
DEFINITIONS AND BASIC FACTS
39
from an affine subspace of Vi to an affine subspace of V2; such a map can of course be extended to an affine map defined in all of Vi. The intersection of any family of affine subspaces is an affine subspace in view of (2.1.3). For every subset EofV the intersection ah(J5^) of all affine subspaces containing E is therefore an affine subspace, the affine hull of E^ and N
(2.1.5)
N
8ih(E) = { ^ XjXj; Y, ^j = 1. Xj eE,
N = l,2,...
}.
Equivalently, if XQ E £", then ah(£') is the sum of XQ and the linear hull oi {x — xo,x e E}. If the affine hull has finite dimension n it is therefore sufficient to take iV = n I 1 in (2.1.5) and fix n H 1 affinely independent elements x i , . . . , Xn^i E E, one of which can be chosen arbitrarily. The following proposition is trivial but important: P r o p o s i t i o n 2.1.2. IfT is an affine map V\ ^ V2 where Vj are vector spaces, and Xj is a convex subset ofVj, then TXi = {Tx;x E Xi} and T~^X2 — {x e Vi\Tx G X2] are convex subsets ofV2 and V\. By definition a subset X of F is convex if and only \iT~^X is an interval for every affine map T \^^V. Thus the definition of convex sets is forced by the second part in Proposition 2.1.2 if convex subsets of R are to be intervals. P r o p o s i t i o n 2.1.3. The intersection of any family X^, a E A, of convex subsets ofV is convex. For every subset EofV the intersection ch.{E) of all convex sets containing E is therefore a convex set, called the convex hull of E. We have N
(2.1.6)
ch(J5) = {YXjXj]\j
N
> 0 , ^Xj
= 1, Xj e E, i V  1 , 2 , . . . } .
Proof. Since {(AQ, . . . , XN) ^ R^"^^; Xj > 0, ^ Q XJ = 1} is a convex set, the set on the right in (2.1.6) is convex for fixed N and XQ^ . . . ^x^. All points obtained from the points XQ, ... ,X]\^ or 2/0? •• • ^VM can also be obtained using the points XQ,. .., xjsf^yo,..., yu together, which proves that the set on the right of (2.1.6) is convex. By (2.1.1)' it is contained in every convex set containing £", which proves the proposition. Note the analogy of (2.1.6) with (2.1.5), and recall that in (2.1.5) it suffices to take N — 1 equal to the dimension of F , or even the dimension of ah(J5). To prove an analogue for convex sets we shall first make some
40
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
geometric observations concerning simplices, which we actually encountered already in the proof of Proposition 2.1.3. The convex hull E oi k } 1 afRnely independent points XQ, .. • ,Xfc G V is called a k simplex. All k simplices are afRnely equivalent. One representation is obtained by taking rro = 0 and Xj = ( 0 , . . . , 1 , . . . , 0) G R^ with 1 at the j t h place only when j = 1 , . . . , A:. Then ( A i , . . . , A^) G E ii and only if Aj > 0, j = 1 , . . . , fc, and X^i Aj < 1. Another more symmetric representation is obtained by taking for Xj the point (AQ, . . . , A^) G R^"^*^ with Aj = 1 and the other coordinates 0. Then E is defined by
A , > 0 , j = 0,...,A:,
5^A,l.
Indirectly we have already encountered these representations several times. In fact, given a standard simplex E in some vector space W, with vertices ^^"05 • • • 5 ^fc5 and points Xj G V there is a unique affine map (f from ah(£') to V such that ^{(TJ) = Xj, jf = 0 , . . . , fc, and
'P(E) = { J ] XjXyAj
> 0, ^
A^ = 1}.
Thus (2.1.1)' means that the affine image of a simplex is contained in X if this is true for the vertices. L e m m a 2.1.4. IfT is an afRne map from the affine hull of the k simplex S with vertices (JQ, . . . ,cr^ ^o an affine space W of dimension < k, then TS = UiTSj where Sj is the k — 1 simplex obtained by omitting the vertex Proof. It is obvious that yJ^TSj C TS. li x e S then L = T'^Tx is an affine space of dimension > 1 which contains a;, and TL = Tx. In the realisation above of S as a compact subset of R^, the boundary dS is equal to UQSJ. If y G L , 2/ 7^ X, then / = {t G R; (1 — t)x \ ty E S} is a compact interval, and if t is in the boundary then (1 — t)x ^ ty E L D dS, which proves that Tx G TdS = UQTSJ. We can take j 7^ 0 here unless the affine line spanned by x and y intersects S only in So But then it would be contained in the affine subspace spanned by So and intersect the boundary dSo of So there, by the (fc — l)dimensional version of the result already proved. Since dSo C U^dSj^ this completes the proof. Combining Lemma 2.1.4 with Proposition 2.1.3 we obtain the following result:
DEFINITIONS AND BASIC FACTS
41
T h e o r e m 2.1.5 ( C a r a t h e o d o r y ) . If E is a subset of a vector space V and the afRne hull W of E has finite dimension n, then n n ch{E) = (^XjXj.^Xj 0
= l,Xj > 0 , Xj GE, i = 0 , . . . , n } .
0
Here XQ can be fixed arbitrarily in E. If E is compact, then ch{E) is also compact. Proof. We have to show that if x is of the form ^ Q XJXJ as in (2.1.6) with N > n^ then x is also of this form with N replaced hy N — 1 and one of the points x i , . . . , x^ omitted. But that is just Lemma 2.1.4 applied to the affine map sending the vertices d o , . . . , crjv of the N simplex to XQ, .. •, xj^. Prom the part of the theorem already proved it follows that ch(£^) is the range of the continuous map n (Ao, . . . , Xn^Xo, . . . , Xn) ^^ / ^ XjXj, 0
defined in the subset of R'^"*"^ x E'^'^^ where Xj > 0, j = 0,... ,n and X^o ^j ~ ^ ^^ ^^ compact HE is compact, so ch{E) is then compact, which proves the theorem. If the intersection of a family of aSine subspaces of a vector space of dimension n is empty, then it is clear that one can find n\l subspaces with an empty intersection, for the dimension must decrease if the intersection decreases. There is an analogue of this too for convex sets; as we shall see later it is closely related to Caratheodory's theorem: T h e o r e m 2.1.6 (Helly). If Xj, j = l , . . . , i V , are convex subsets of a vector space V of dimension n, and if Xi^ fl • • • fl Xi^ ^ 0 for arbitrary z'o,..., 2n ^ {I5 • • • 5 ^}j then H^Xj ^ 0. This remains true for Xj, j E J, when J has arbitrary cardinality, if all Xj are closed and some finite intersection is compact. Proof. By a standard definition of compactness the second part of the statement follows from the first, which will be proved by induction with respect to N. Thus assume that iV > n 4 1 and that the statement has already been proved with N replaced by iV — 1. Then we can for i = 1,... ,N find Xi e r\i<j n + 1 these points are aSinely dependent, so we can find ( A i , . . . , A^v) 7^ 0 so that N
Y^ XiXi = 0,
N
^Xi
= 0.
42
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Write Xi = Xf  A" where Xf = max(±Ai,0).
Then A = E f \ ^
=
E f A > 0, and N
N
N
N
^ = J2iK/^)^i = E(VA)^^, E(^^^A) = E(v/A) = 1We have x G H^Xj. In fact, if A^ > 0 then A~ = 0, and since Xi E Xj when i ^ 3^ the second representation of x proves that x £ Xj, in view of the convexity of Xj. Similarly, using the first representation, we find that X E Xj if A^ < 0, and all together this proves that x E HJ^Xj, so the intersection is not empty. E x e r c i s e 2.1.2. Write down explicitly what Kelly's theorem states when n = 1, and give a direct proof. Prove that if Xj C R"^ are intervals, that is, products of intervals in the different coordinates, then d^Xj ^ 0 if Xj n Xfc 7^ 0 for arbitrary j , k < N. We can give Theorem 2.1.6 a more precise form if we make the following definition: D e f i n i t i o n 2.1.7. By the dimension of a convex subset £" of a (finitedimensional) vector space V we shall mean the dimension of the affine hull Sih.{E), that is, the supremum of all n such that there exists an affinely independent (n f l)tuple XQ,. .. ,Xn E E. Note that the simplex with vertices XQ,. .. ,Xn has interior points as a subset of the affine hull W of E/ii XQ,. .. ,Xn is an affine basis for W. The interior (boundary) of E considered as a subset of W will be called the relative interior (boundary) of E. The relative interior is dense in E, for it contains some point y, and if x is any point in E, then Xx \ {1 — X)y is in the relative interior when 0 < A < 1. E x e r c i s e 2.1.3. Show that the convex set E is relatively open if and only if every affine line intersects E in a. point or an open interval. E x e r c i s e 2.1.4. Show that Theorem 2.1.6 holds if n is replaced by 1 + m i n d i m X j or by dimch(UXj). We have seen that every point x in a fe simplex has a unique representation of the form x = ^ Q Xjaj where aj are the vertices and Xj > 0, 5^0 A,  1. We shall now discuss to what extent a similar representation holds for an arbitrary convex compact set K in a. finitedimensional vector space V. It is clear that such representations cannot avoid using points which are extreme in the sense of the following definition:
DEFINITIONS AND BASIC FACTS
43
Definition 2.1.8. A point x in the convex set K is called extreme if 2
X = A i ^ i + A2a:2, Xj G K , Xj > 0, j = 1, 2, Y"] Aj = 1 1
An apparently stronger condition is that N
x = ' ^ XjXj,
N
Xj e K,
Xj >0,
j = 1,...,N,
^
1
Xj = 1 1
—^
X J ^= x^ J = i , . . . , i V.
The equivalence follows as usual by induction for increasing N. It is obvious that the extreme points of ch(j5') must belong to E \i E is an arbitrary set. E x e r c i s e 2.1.5. Determine ch(£'i U E2) and the extreme points of ch(£;i [JE2) where Ei  {(xi,X2,0);x? + x  = 1} and E2 = {(1,0, ± 1 } . (Note that the set of extreme points is not closed!) T h e o r e m 2.1.9 (Minkowski). IfK is a compact convex set of dimension n in a Gnitedimensional vector space, then every point x E K can be written in the form n
(2.1.7)
n
X = Y^XjXj,
Xj > 0, j = 0 , . . . , n , ^ A ^  = 1,
0
0
where all Xj are extreme points of K. The representation all X in the relative interior of K unless K is a simplex.
is not unique for
Proof. The existence of a representation (2.1.7) is obvious if n = 1, for K is then an interval and the extreme points are the end points of the interval. For n > 1 the proof will be given later by induction with respect to n, after we have proved the required separation theorem. However, the lack of uniqueness is easily proved at this time. If K is not a simplex then there are at least n\ 2 different extreme points XQ,. .. ^x^^i These must be afBnely dependent, so we have n+l
n+1
0
0
where not all JJLJ are equal to 0. Then x = ^ Q ^ J / ( ^ + 2) can also be written a: = ^ Q Xj(He/Xj)/(n + 2), and all the weights (1f £/Xj)/(n + 2)
44
IL CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
are positive if e is small enough, but not for all £ > 0 nor for all £ < 0. Hence we can find two representations of x as center of gravity of only n f 1 points among XQ, . . . , Xn\iRemark. Theorem 2.1.9 has an infinitedimensional analogue called the KreinMilman theorem. It states that a compact convex set is the closed convex hull of its extreme points. Refinements due to Choquet [2] state that it is enough to take averages of extreme points weighted by positive measures, and he has also proved an analogue of the last part of Theorem 2.1.9 with an appropriate definition of simplices. We shall now discuss separation theorems. We shall refer to them as the HahnBanach theorem although that is only appropriate in the infinitedimensional case. T h e o r e m 2.1.10. Let X be a finitedimensional vector space hyperplane W such that XQ ^ W function f in V with f{xo) = 0
0, then tiXi + t2X2 = (ti + t2){{ti/{ti + t2))Xi + (t2/(ti + t2))x2) G X. If X G X then —x ^ X. Since V \ {0} is connected we can find a point y ^ 0 in the boundary of X , and y ^ X because X is open. In every neighborhood of y we can find x G X , hence —x ^ X , which proves that —y ^ X. Hence X f) Ky = 0, which proves the theorem when d i m F = 2. (b) If dim V > 2 we let W he a. subspace of maximal dimension with W^ n X = 0. We have to prove that W is a hyperplane, that is, that dim{V/W) = 1. Let T : V —^ V/W be the canonical map, assigning to an element in V its residue class mod W. Then T X is convex (by Proposition 2.1.2) and relatively open (see Exercise 2.1.3). If diin{V/W) > 1 we take a twodimensional subspace H and note that H fl T X is also convex and relatively open. By part (a) of the proof there is a line Hi C H with HiHTX = 0, which means that ( T  ^ i f i ) H X = 0. Since T^Hi D W this contradicts the maximality of W and completes the proof.
DEFINITIONS AND BASIC FACTS
45
Corollary 2.1.11. Let X be a convex and closed subset of a finitedimensional vector space V. If XQ ^ X there is an afHne hyperplane containing XQ which does not intersect X, that is, there is an affine function f with f{xo) < 0 < f{x), xeX. Proof. Let U be an open convex set such that XQ e U and [/ D X = 0. Then Y = {xy;xeX,yeU}= \J{xy]yeU} xex is open and convex, and 0 ^ Y. By Theorem 2.1.10 we can find a linear form L such that L > 0 in y , that is, L{x) > L{y) ii x e X and y ^ U. Now sup L{y) = L{xo) + sup L{y  XQ) = L{xo) + 6 yeu yeu where ^ > 0, so f{x) = L{x) ~ L{XQ) — 6 has the required properties. The following statement is closer to Theorem 2.1.10: Corollary 2.1.12. IfX is a closed convex subset of a Gnitedimensional vector space V, and ify is on the boundary of X, then one can find a nonconstant affine f such that f{y) = 0 < f{x), x E X. The affine hyperplane {x E V\ f(x) = 0} is called a supporting plane of X. Proof. The statement is trivial if d i m X < d i m F , so we may assume that the interior X° of X is not empty. By Theorem 2.1.10 we can choose / so that f{y) = 0 < f{x), x G X ° , which implies that 0 < f{x), x eX. Corollary 2.1.13. An open (closed) convex set K in a finitedimensional vector space is the intersection of the open (closed) half spaces containing it. Proof. This is an immediate consequence of Theorem 2.1.10 and Corollary 2.1.11. Remark. The infinitedimensional analogue of Theorem 2.1.10 and its corollaries is the HahnBanach theorem which can be found in every text on functional analysis. E x e r c i s e 2.1.6. Prove, with the notation in Corollary 2.1.12, that if V = BJ^ with the Euclidean norm  •  and scalar product (•,•), and if X is compact, then {xyj')t\f'\ > 0.
Conclude that X is the closed convex hull of the set of all x G dX such that there is a closed ball B D X with x G dB; such points are extreme.
46
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
(Hint: Assume that this is not true and derive a contradiction by taking t large in the preceding statement.) End of proof of Theorem 2.1.9. It is no restriction to assume that K has interior points. If x G dK^ it follows from Corollary 2.1.12 that we can find an affine function / with / > 0 in if and f{x) = 0. The intersection Ki = {y E K; f{y) = 0} is convex and compact, x G i^i, and dim if i < n. It is clear that an extreme point of if i must be an extreme point of K. If the theorem has been proved for lower dimensions we can therefore write n —1
(2.1.8)
n —1
X = Y ^ AjXj,
where Aj > 0, Xj G i f i , 0 < j < n, Y ^ A^ = 1,
0
0
and Xj are extreme points of i f i , hence of if, which is better than the claim for a general point. In particular, we conclude that extreme points exist. If X is in the interior of if we choose an extreme point Xn and note that the intersection of if with the line through x and Xn is an interval with end points Xn and rr' G OK. Hence x' is of the form (2.1.8), and since X = fiXn + (1 — f^)x^ for some fi G (0,1), we obtain nl
X = y . ^ji^ ~ f^)^j + M^n5 0
which completes the proof. We shall now discuss an example. An nxn matrix {ajk) is called doubly stochastic if the elements are nonnegative and the row and column sums are equal to 1, djk > 0, j,fc = l , . . . , n ; ^
a^fc = 1, fc = 1,. • •, n; ^
j=i
a^fc = 1, j = 1 , . . . , n.
k=i
Important examples are matrices where all the elements are equal to 0 or 1, which means that each row and each column contains precisely one element equal to 1. Thus there is a permutation a of { 1 , . . . , n} such that ajk = 1 if k = aj and ajk = 0 ii k ^ aj. One calls (ajk) a permutation matrix then, fo^ Yl'k = l^jk^k = ^ajT h e o r e m 2.1.14 (G. BirkhofF). The nxn doubly stochastic matrices form a compact convex set Dn of dimension (n — 1)^, and the extreme points are the permutation matrices. Thus every doubly stochastic matrix (ajk) is of the form ^jk = X ^ ^^^ k=aj
where A^ > 0, ^
A^ = 1,
DEFINITIONS AND BASIC FACTS
47
and a runs over the group 6 ^ of permutations of { 1 , . . . , n } . One can take A^ = 0 except for at most (n — 1)^ + 1 permutations. Proof. (2.1.9) defines Dn as the intersection of closed half spaces and 2
affine hyperplanes in R'^ , so it is clear that Dn is closed and convex. Since 0 < ajk < 1 it is also clear that D^ is bounded, hence compact. The equations in (2.1.9) mean that (Ink = l  ^ J ^  ^ ' ^ ' ^ < '^5 o.jn = l  X / ^  ^ ' ^ ' ^ ^ '^' ^^^ ^ 2  n + 22 j Q \i Xj > 0, j — 1 , . . . , n, and we shall write ax = ( x ^ i , . . . , x^^) if cr G Sn If 0 < a G R"^ we shall write x^ — Y[^ ^V • T h e o r e m 2.1.15 (Muirhead). Ifa,/3e the following conditions are equivalent:
IC" and a > 0,/3 > 0, then
(i) There is a constant C such that
(2.1.10)
Z l ^""^  ^ 5 ] ^'''''
0 < X G R^;
48
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
(ii) the inequality (2.1.10) holds with C = 1; (iii) P is contained in the convex hull of {crof; a G S n } ; (iv) ifa"" and ^* are the decreasing rearrangements of a and (3, then
^/3;<x:«;, i an when a E A. E x e r c i s e 2.1.8. Determine the extreme points in the set of n x m matrices (ajk) such that (2.1.9)''
ajk>0,
i =
l,...,n,fcl,...,m;
n
m
^ajk
4Onj sin6>,
0{6)n,j =
Ok\i,jsin6\On,jCos6,
the other elements being the same as in O. Then
o{e)^^\ai,. . . , < ) = (A,.. .,Pk,aie),pk+i, •. .,pn2,bie)) where a(0) = a and a(7r) = b. Hence we can choose 0 so that a{6) = 13^1 and 0 ( ^ ) ^ ( a i , . . . , a * ) is then a permutation of /?, for h{6) = /J^ since a{0) f b{0) — Pni + f^n for every 6. This completes the proof. Exercise 2.1.9. Prove that the diagonals /? of the Hermitian symmetric n X n matrices H = (Hjk) with prescribed eigenvalues ( a i , . . . , a^) G R^ form a convex set described by Theorem 2.1.15. Conclude using Exercise 2.1.7 that if / is a convex symmetric function in R^ then H H> / ( a i , . . . , o^n) is convex in the n^dimensional space of Hermitian symmetric matrices. (Hint: Prove that n
j,k
= l
n
j,k
= l
where [/ = (C/j/^) is a unitary matrix.) Our next example of convex sets occurring in analysis concerns the numerical range of an operator. T h e o r e m 2.1.18 (HausdorfF). Let V be a (Rnitedimensional) vector space over R with a Euclidean scalar product denoted by E{, ), and let Ti, T2 be two linear transformations V ^ V. Then the numerical range O = {{E{Tix,x),E{T2X,x))]x
G V,E{x,x)
= 1} C R^
is convex if dim V > 2, and if dim V = 2 it is an ellipse, possibly to an interval or a point. Proof. Let 61,62 be two orthonormal vectors, that is, E ( e i , e i ) = ^(62,62) = 1, £"(61,62) = 0,
degenerated
52
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
and write Qj{x,y) = \{E{TjX,y) •^E{x,Tjy)), which is a symmetric bilinear form. Then e{9) = cos^ei + sin0e2 is a unit vector for every ^ G R, so f7 3 ( Q i ( e ( e ) , e ( ^ ) ) , Q 2 ( e W , e ( ^ ) ) ) . Since 2cos2e = 1 + cos 20, 2sin^0 = 1  cos26>, 2 sin l9 cos 0 = sin2 0 (if D < 0), when 6 goes from 0 to TT. When d i m F = 2 the proof is now complete. When d i m F = 3 we can choose a third unit vector 63, orthogonal to ei and 62, and continuously deform 61,62,63 to 62,61,63 by changing 61,62,63 to  6 3 , 62, 61 to 62,63,61 to 62, 61,  6 3 by rotations around one of the vectors. Let the corresponding vectors be 6^, 0 < t < 3. Then the range i?(6*,e2) must for some t G [0,3] pass through an arbitrary point y in the interior of ^(61,62) — i?(6i,62) = i?(6i,62), for the winding number of the oriented ellipse around y changes sign as t goes from 0 to 3, and it is a continuous function of t when y ^ i?(6i, 62). Hence Q contains the interior of i?(6i, 62) also. Since two arbitrary points in V belong to some twodimensional plane, the convexity of Q follows. C o r o l l a r y 2.1.19. Let V be a Gnitedimensional complex vector space with positive definite Hermitian scalar product denoted by E, and let T be a complex linear transformation in V. Then the numerical range n = {E{Tx,xy,x is a convex subset
e V,E{x,x)
= 1}
ofC
Proof. If V has complex dimension one then T is just multiplication by a constant, and the statement is true. If the complex dimension of ^ is > 1, we can regard I^ as a real vector space of dimension > 4, with Euclidean form Re E. If we write T = Ti \ iT2 where Tj are Hermitian symmetric, then E{Tx, x) = E{Tix, x) + iE{T2X, x) = Re E{Tix, x) + i Re E{T2X, x), and the statement follows from Theorem 2.1.18. It is the corollary which is usually referred to as the theorem on the numerical range.
DEFINITIONS AND BASIC FACTS
53
Exercise 2.1.10. Prove that with the notation in Corollary 2.1.19, the numerical range of T contains the convex hull of the spectrum of T, that is, the set of 2: E C such that T — z Id is not invertible if Id is the identity operator, and that there is equality if T is normal. Exercise 2.1.11. Prove that the numerical range of a doubly stochastic nxn matrix acting on C^ with the standard Hermitian metric is contained in the convex hull of the roots of unity e^^*^/^ where 0 < z/ < /x < n. Theorem 2.1.14 studied a special convex polyhedron: Definition 2.1.20. The convex hull of a finite set E C V is called a convex polyhedron. It is clear from the definition that a polyhedron is compact and that the extreme points form a subset of E\ hence it is finite. T h e o r e m 2.1.21. A convex polyhedron X CV is the intersection of a, finite number of closed half spaces H.IfX has interior points, they can be chosen so that X fl dH has interior points relative to dH. Conversely, any bounded intersection of a finite number of closed half spaces is a convex polyhedron. Proof. Let ^ be a finite subset of V with ah(£J) = V^ so that the polyhedron ch(jE') has interior points. If a: G dch{E) then there is a half space H D ch{E) such that (dH) H ch{E) contains x and dim{{dH) D ch{E)) = n — 1 where n — d i m F . This is obvious when n = 2. A general proof can be made quite parallel to that of Theorem 2.1.10. Let x = 0^ and choose H D E containing 0 so that i^ = dim(W fl ch{E)), W = 9 i ? , is as large as possible. If z/ < n — 1 we let W be the linear hull oiWr]ch{E) and observe that the image K of ch{E) in V/W, which has dimension n — i/ > 2, is also a convex polyhedron with 0 on the boundary. From the (n — z^)dimensional case it follows that there is a half space Hi in V/W containing K with 0 E dHi and diin{K fl dHi) = n — u — 1 > 1. Then the inverse image H2 of Hi in V contains ch(£^), and (dH2) fl ch(E) contains W f) ch.{E) as well as points not in W, so the dimension is > z/. This contradiction proves our claim. Now an afline hyperplane is uniquely determined by n afiinely independent points, so it follows that ch{E) is the intersection of at most (dimv) half spaces, where \E\ is the number of points in E. Conversely, let X be the intersection of a finite number of closed half spaces i J j , j = 1 , . . . , J , and assume that X is bounded. If x is an extreme point of X , then it follows (as in the proof of Theorem 2.1.14) that X must be in n = dim V aSine hyperplanes dHj intersecting only at x, which proves that there are at most {^^^y) extreme points. By Theorem 2.1.9 X is the convex hull of this finite set.
54
11. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
We shall now prove some simple facts on convex functions which are analogues of results proved above for convex sets. We label them to suggest the connection. T h e o r e m 2.1.2'. Let T be an affine map V\ —> V2 where Vj are vector spaces, let Xj be convex subsets ofVj, and let fj be a convex function in Xj with finite values. Then Fi{x) = f2iTx),
X G T'X2,
F2{y) = inf h{x), y G TXx, Tx=y are convex functions if F2{y) > —00 for some y in the relative interior of TXi. Proof. The statement on Fi is an immediate consequence of Proposition 2.1.2 or (2.1.2). To prove the one on F2 let yi,y2 G TXi and Ai,A2 > 0, Ai 4 A2 = 1. For arbitrary Xj G Xi with Txj = yj we have F2{Xiyi + A2y2) < /i(AiXi + A2X2) < Ai/i(a:i) + A2/i(x2). Taking the infimum over all permissible Xj we obtain F2{Xiyi + X2y2) ^ XiF2{yi)hX2F2{y2) Ifi^2(2/i) =  o c it follows that F2(Ai?/i + A22/2) =  0 0 for all ?/2 G T X i , so F2 would be equal to —00 in the relative interior of T X i , which is against the hypothesis. The proof is complete. T h e o r e m 2.1.3'. If X is a convex set and fa, a E A, is any family of convex functions in X with values in (—00, Hoo], then f{x) — sup^,^^ / a ( ^ ) is also convex; it is the smallest convex majorant of all f^ If info, ^ A / a ( ^ ) < 00 for every x E X, then the largest convex minorant is g{x) = i n f { ^ Xafa{Xa)]Xa
>0,XaeX,
^
Ac, = 1, ^
XaXa = x},
provided that g{x) > — 00 for some x in the relative interior of X. In the definition of g it is assumed that only a finite number of X^ are ^ 0, and it does not change if we require that at most 1 + d i m X of them are / 0. Proof. The statement about / is a very easy exercise. The convex hull of the epigraphs of the f^ is defined by t > g{x) with equality excluded if the infimum in the definition of g is not attained. We may assume that X has interior points. Hence it follows from Caratheodory's theorem 2.1.5 that it sufiices to take 2 + d i m X of the AQ, nonzero in the definition of g. In taking the convex hull over the point x we may let one of the points be (/a(^) + 1?^) fo^ some a such that fa{x) < 00. But a positive weight for this point would increase the infimum, which completes the proof. (That g{x) > —00 everywhere follows as in the proof of Theorem 2.1.2'.) The results on diff'erentiability of convex functions proved in Section 1.1 can easily be extended to several variables:
DEFINITIONS AND BASIC FACTS
55
T h e o r e m 2.1.22. Let f be convex and finite in an open convex subset X of a Gnitedimensional vector space V. Then f is locally Lipschitz continuous, f'{x;y)= Ihn (fix+ hy)fix))/h exists for every x £ X and y E V, and the limit is uniform in y when y is bounded. The Gateau differential (or subdifferential) f'{x]y) is convex and positively homogeneous, f{x',ty) (2.1.12)
= tf{xy),
t>Q,yeV,
f ( x ; y i + y 2 ) < f ( : r ; ^ i ) + /'(x;7/2), f'{xy) 0; x/t e B} = 1/ sup{t; tx G B}
is a convex positiirely homogeneous the origin, that is, F{x)>0 (2.1.14)
function,
ifO^xeV,
which is positive except at
F{0)=0]
F{tx) = tF{x),
t > 0,
xeV
F{x + y) 0 such that dx^B is locally concave in the subset of XR = {rr G X ; x < R} where dx,B < 2^^. Then f{x) = max{dx,B{x),
6R{\X\
 R)/R),
x G
XR,
is locally convex in XR, the supremum is 0 in every component, and when t < 0 then {x G XR\f{x) < t} is compact since it stays away from the boundary of X and from {rr; x = R}. Hence it follows from Theorem 2.1.25 that every component of XR is convex, and when i? —> 00 it follows that X is convex. Thus the first three conditions are equivalent, and by
60
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Theorem 2.1.25 we also have (v) => (i). On the other hand, if X is convex then is convex in X (see Exercise 2.1.13) and tends to +oo at the boundary and infinity, so (iv) is fulfilled. The proof is complete since (iv) = > (v). Note that (iii) is a local condition on X at dX: If X is locally convex in the sense that every point in dX has a (convex) neighborhood U such that C / n X is convex, then it follows that X is convex (if connected). This local property is also evident from the following: T h e o r e m 2.1.27. Let X he an open connected set in R"^ which is not convex. Then there is a point z G dX and a quadratic polynomial q with q{z) — 0, ^'(/2^) 7^ 0, such that q{x) < 0 implies xEXifx is close to z, and {t,d)q{z) = 0,
{t,dfq{z) eh^^
\h\ < 6,
so when x = ZQ the minimum in (2.1.18) is a nondegenerate minimum achieved Sit h = 0. By the implicit function theorem it follows for x near ZQ that the minimum is still a nondegenerate minimum taken at a point h which is a C^ function of x. Hence (/? is a C^ function in a neighborhood of ZQ, ^'{ZQ) = i^, and since (p{zo + ht) < —eh'^ it follows that {t^d)(p = 0 and that {t^d)^cp < —2e at ZQ. For the quadratic polynomial
q{x) = Yl ^ X ^ o ) ( ^  zor/a\
+ rj\x  zo\'
a cp{x) if rj > 0 and x — ZQI is small, so q{x) < 0 implies ip{x) < 0, hence x ^ X. If r/ is sufRciently small we have {t,d)^q{zo) < 0. The proof is complete. Corollary 2.1.28. An open connected set X C R'^ with C^ is convex if and only if the principal curvatures are > 0 at every point.
boundary boundary
We shall finally discuss another characterization of convex sets in terms of a distance function, now in the exterior of the set. We begin with a general fact about distance functions, for the sake of simplicity stated only for the Euclidean distance in R^. L e m m a 2.1.29. Let F be a closed set in IV^ and set f{x) = minx — z\'^ where \'\ is the Euclidean norm. Then we have f{x \y) = f{x) + f'{x] y) + o(y), f'{x\y)
7/ > 0,
= min{(2y,x  z);z e F,\x  z\^ =
where f{x)}.
This is Lemma 8.5.12 in ALPDO, and we refer to the proof given there. Note that the Gateau differential f'{x]y) is a concave, positively homogeneous function of y just as it would have been if / had been a concave
62
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
function. If dpix) = \/f{x) is the distance from x to F , it follows from Lemma 2.1.29 that dp is Gateau differentiable at every point x ^ F , with differential y H^ min{(y, x  z)j\x  z\\ z^F,\x
z\ =
dpix)},
and that dp is differentiable at x if and only if there is a unique point in F at minimal distance from x. This is used in the statement of the next theorem: T h e o r e m 2.1.30 ( M o t z k i n ) . Let F be a closed set in R^. Then F is convex if and only if the Euclidean distance function dp is differentiable at every point in ZF, or equivalently for every x ^ F there is a unique point in F at minimal distance from x. Proof. If F is convex and z ^ F has minimal distance from x G C F , then {y — z,x — z) < 0 for every y E F, for
\xzf
< \xze{yz)\'^
= \xz\^2e{xz,yz)+e^\yz\^,
0 < 6 < 1.
(Note that this gives another proof of Corollary 2.1.11.) Taking e = 1 we conclude that x — >2^ < x — ?/ if F 3 y ^ z. The condition in the theorem is therefore necessary. Assume now that F is not convex. We must then prove that there is an open ball B with B H F = 9 such that B H F contains more than one point. That F is not convex means that we can find an interval [xi, 0:2] with xij^2 ^ F , xi 7^ X2, such that the interior is in C F . We place the origin at the midpoint so that X2 = —xi, and write B{w^r) = {x; x — it; < r } , w eIC',r>0. Choose ^ > 0 such that 5 ( 0 , ^ ) D F := 0. If B{w, r) D 5 ( 0 ,
Q),
Xj i B{w, r), j = 1,2,
then r > t(; + g,
\w ± x i p > r^,
hence \w\'^ +  x i p > r^ > {\w\ + ^)^, so (2.1.19)
\W\ 0, so x G T^ \ip{x) 7^ 0. Hence VQ is open and closed in {x G V]p{x) ^ 0}. Now TQ is connected, for if X G r^, then x^t0 eVe when t > 0, hence Ax + /x0 G Tg for all A > 0 and // > 0. This proves that TQ is even starshaped with respect to 0, and
that r^ = r. If 2/ G r and ^ > 0 is fixed, then Ey^e = [x eV] p{x + iE0 + isy) 7^ 0, Re s > 0} is open, and 0 G Ey^^ since p{ie0 + isy) = {is)^p{£0/s + y) = 0 implies 5 < 0. If X G Ey^e, then p{x + ie0 + isy) 7^ 0 by Hurwitz' theorem if
64
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Re5 > 0, and this remains true when R e s = 0 since x + isy is real then. Hence Ey^e is both open and closed, so Ey^e — V Thus p{x + i{ee + y)) ^ 0 ,
if X G R^, y G r , £ > 0.
Since F is open, this remains true when ^ == 0, so the equation p(x + ^y) = 0 has only real roots, for ii t = ti ^ it2 is a root with ^2 7^ 0 we would get p{{x + tiy)/t2 4 iy) = 0. Hence y can play the role of ^, so 7 is starshaped with respect to every point in F, hence convex. Finally we shall prove that if 7/ G F and x G V, then 5 1—> {p{sx + y)Y^'^ is concave when sx \y ^V. With t^ G R we have m
p{x+ty)=p{y)Y[{tti),
m
thus p{sx+y) = s'^pixhy/s) =
p{y)Y[{lsti),
and sx i y ET means that 1 — st^ > 0 for every i. If f{s) = logp{sx + y) then
hence
by CauchySchwarz' inequality, which proves that s f^ p{sx + y)^^'^ concave. The proof is complete.
is
E x a m p l e . The simplest example, and the one motivating the name, is the hyperbolic quadratic form p{x) ^=^ x\ — x\ —  • • — x^ in R^. The convexity of F just means the convexity of the forward light cone, defined by xi > \/x\ + • • • 4 x^, and the concavity of yjp means that we have the reversed triangle inequality \/p[x + y) > \/p{x) + \/p{y) for all x, ?/ in the forward light cone. Other important examples are the space of n x n real symmetric matrices (or hermitian symmetric matrices), with p{x) — d e t x and 6 equal to the identity matrix. The cone F consists of the positive definite matrices then. If X G F and y EV^ then the fact that t \^ p{x + ty) has only real zeros implies that this is also true for the derivative with respect to t; if x G F then all the zeros are negative. The derivative is equal to mq{x{ty) where q[x) = p{y^x^... ,x) with p denoting the polarization of p. (See Appendix
DEFINITIONS AND BASIC FACTS
65
A.) Hence q is also hyperbolic with respect to every vector in F. If ^(^) > 0 then p > 0 and g > 0 in F. By the concavity of p^/"^ it follows according to Proposition A.l, condition (iv), that
Since q satisfies the same hypotheses as p^ with F replaced by a cone containing F, we can prove by induction: Corollary 2.1.32. If the hypotheses and p{6) > 0, then (2.1.21)
p{xu ...,xm)>
of Proposition
V^XiY'"^ • "V{xmf''^,
2.1.31 are fulhlled
X i , . . . , x ^ E F.
Ifp is complete in the sense that p{x + ty) = p{x) for all x^ t impUes y = 0, then X H^ p{x, ...,x, yn+i, • • •, Um) is complete ifn > 2 and 2/n+i, ".ym ^ F. In particular, this is a nondegenerate quadratic form if n = 2, and p ( x i , . . . , Xm) > 0 ifxi G F \ 0 and Xj G F when j = 2 , . . . , m. Proof. Taking y = xi and assuming that (2.1.21) is already proved for hyperbolic polynomials of degree m —1, we obtain with q{x) = p{y^ x , . . . , x)
which proves (2.1.21). Alternatively we could use that X^p{x,X,X^,...,Xm)
is a hyperbolic quadratic form with forward light cone containing F, if 3^3,... ,Xm G F. This implies the condition (A.7) with f = p (see Remark 2 after Proposition A.2), and hence the inequalities (A.8)(A.10) follow. To prove the last statement it suffices to show that q is complete iim, > 3 and q is defined as above. Suppose that q{x + tz) = q{x) for all x and t. In particular, q{y^tz) = q{y)^ that is, q{ty\z) — q(ty)^ sop{ty\z)—p{ty) = a is independent of t. Since the zeros of p{ty) + a = t^p{y) + d must all be real, it follows that a = 0. Thus p{y h sz) — p{y) ^ 0 for all 5, so it follows that y + 52; G F for every s. Hence (sx 4 y + sz)/{s + 1) G F,
if x G F, 5 > 0,
and letting s —^ oo we conclude that j:f >2: G F for all x G F, hence xhz ET then. We can replace z by tz for any t, so x \ tz E V for all t and x G F. Thus p{z 4 sx) cannot have any zeros ^ 0, so p{z + sx) = s'^p{x)^ that is, p{x + tz) — p{x) for all t and all x G F. But this implies 2; = 0 since p is complete.
66
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Exercise 2.1.14. Prove that ii p{z) = Yl^ ^j^^ i^ a polynomial in z G C, then the zeros o{p'{z) are either zeros of ^(z) or else contained in the relative interior of the convex hull of the zeros oi p{z). (Hint: Study p'{z)lp{z).) Exercise 2.1.15. Let if be a convex subset of C, and let ^{z) = Y^^ djZ^ be a polynomial in z G C. Prove that the set of all w E C such that all the zeros of p{z) = w are contained in K is a convex set. (Hint: Apply the preceding exercise to a product {p{z) — wi)'^^{p{z) — W2)'^'^) 2.2. T h e Legendre transformation. In Definition 2.1.1 we introduced the notion of convex function, with values in (—oc, cx)], defined in a convex subset X of the vector space V. It is clear from the definition that the function remains convex if we extend it to V by defining / = hoo in V \X. For the sake of convenience we shall always assume in this section that the convex functions considered are defined in all of V. P r o p o s i t i o n 2.2.1. If f is a convex function in V then X = {xeVf{x)
0, for it is clearly convex on a line where x is constant, and d? f{x^ ax + b)/dx'^ = 2b'^/x^ > 0. The limit of f{x,y) as (x^y) ^ 0 along a ray is 0, but the limit along the parabola x — ay'^ is 1/a. The best way out of the problem is to make / lower semicontinuous: P r o p o s i t i o n 2.2.2. Let f be a convex function in V, and set fi{x)=
lim/(y),
xeV.
Then / i is convex and fi{x) < f{x) for all x, with equality if x is in the interior of X = {x e V]f{x) < oo} in ah(X) or interior in V \X. The function / i is lower semicontinuous and is called the lower semicontinuous regularization of f. Proof. We just have to verify that fi{x) > —oo for every x, and we may then assume that X has an interior point XQ. But then f{x) — / ( X Q ) is
THE LEGENDRE TRANSFORMATION
67
everywhere bounded from below by the Gateau differential f'{xo,x (cf. Theorem 2.1.22).
— XQ)
In the example above we can make / convex in V by defining f{x, y) = joo when a; < 0, but the lower semicontinuous regularization is equal to 0 at the origin. The reason for the importance of lower semicontinuity is that while the epigraph
{{x,t)
ev^iixev,t>f{x)]
is convex if and only if / is convex, it is dosed if and only if / is lower semicontinuous. This will be a decisive point when we extend the notion of Legendre transformation. Denote by V the dual space of V^ and let V xV' 3 ( ^ , 0 ^~^ (^^0 be the bilinear form defining the duality. Definition 2.2.3. Let / be a convex lower semicontinuous function in V^ not identically joo. Then the Legendre transform ( = conjugate function = Fenchel transform) / of / is defined by (2.2.1)
/(0sup((a;,0/(^)),
i ^ V .
xev
As the supremum of a family of affine, hence continuous, functions on V it is clear that / is convex and lower semicontinuous (cf. Theorem 2.1.3'), and we have an inversion formula: T h e o r e m 2.2.4. If f is a convex lower semicontinuous function in V, not identically joo, then the Legendre transform f has the same properties in V , and (2.2.2)
f{x)=snp{{x,0f{0)
Proof. From (2.2.1) we know that f{x) + f{() > (x,^), hence f{x) > snp^^Y,{{x,() — / ( O ) . To prove the opposite inequality we choose any c G R with c < f{x). Since (x, c) is not in the closed convex epigraph of / , it follows from Corollary 2.1.11 that we can find ^ G V^', & G R and ^ > 0 such that (2.2.3)
( x , 0 ^ c b > {y,0+tb^£.
iiyeV,t>
f{y).
Assume at first that f{x) < oc. When y = x it follows from (2.2.3) that 6 < 0, and dividing by \b\ we obtain with rj = ^/\b\ {x,rj)c>{y,r])t
ifyeV,
t>f{y).
68
11. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Hence /(r/) < (x,?/)  c, that is, c < {x.rj)  f{rj) < f{x), or f{x) which proves (2.2.2) when f{x) < oo. In particular, we conclude not identically Hoo. Now assume that f{x) = +oo. Since f{y) some y eV/it follows from (2.2.3) that 6 < 0. If 6 < 0 we obtain
> /(x), that / is < oo for as before
that f{x) > c, so it remains to examine the case where b = 0. Then it follows from (2.2.3) that (x,^) > ( y , 0 + ^ if f{y) < oo, so it follows from (2.2.1) that f{v + to < fiv) + t{{x,0
e),
iit>0,rje
V.
Hence / > ) > {x, 7/ + to  t{{x, 0~e)
hv)
= {x, v)+t£
m ,
t>0,7,€V'.
If we choose rj so that f{rj) < oo and let t —> +oo, it follows that f{x) +00, which completes the proof.
=
Exercise 2.2.1. Show that the Legendre transform (i) of / f C is / — C, if C is a constant; (ii) o f / ( .  a ) i s / + ( a , . ) , i f a E y ; (iii) o f / ( ^ ) i s / ( . / ^ ) , i f t > 0 ; (iv) o f t / is tf{/t) iit>0. Show that if / = +00 in the complement of a linear subspace Vi oiV, then / is the pullback of the Legendre transform of the restriction of / to Vi by the natural map V ^ V^, with kernel equal to the annihilator of Vi in V. T h e o r e m 2.2.5. Let / i , . . . , /iv be lower semicontinuous convex functions in V such that f = ^^ fj ^ +oo. Then f is the lower semicontinuous regularization of the convex function (infimal convolution)
(2.2.4)
S{0 = jni
^E/.fe).
Proof. Let f{x) < oo. Then ^( —oo for every ^, for fj{x) j = 1 , . . . ,iV, hence
< oo,
It is clear that g is convex, and it follows from (2.2.4) and the inversion formula (2.2.2) for fj that N
sup((a;,0  9(0) = s u p E ( { x , e , )  fjiQ)
N
= 5;/,(a;) =
f{x).
THE LEGENDRE TRANSFORMATION
69
If ^1 is the lower semicontinuous regularization of ^, it follows that gi > f, hence gi < f hy Theorem 2.2.4, and since gi > f we obtain gi = f. The preceding result simplifies if miiij fj{^) < oo for every ^, for g is then finite everywhere, hence continuous, so f = g. In particular, we obtain when iV = 2 by taking the value at the origin: Corollary 2.2.6. Let / i and /2 be convex lower semicontinuous functions in V with / i + /2 ^ +oc, and assume that /2 < f oo everywhere. Then we have (2.2.5)
inf (/i(x) + ^(a;)) + inf ( / i ( 0 + / 2 (  0 ) = 0
We shall give an application of (2.2.5) later on, but we continue now with an analogue of Theorem 2.2.5: T h e o r e m 2.2.7. Let fc^, a ^ A, be convex and lower semicontinuous. Then f = sup^,^^ fa has the same property. Iff ^ hoo then f is the lower semicontinuous regularization of the largest convex minorant g of{fa},
where X^ > 0,a ^ A, only Gnitely many AQ, are ^ 0, and Yl^eA ^Q; = 1The proof is so close to that of Theorem 2.2.5 that we leave it as an exercise for the reader. If iT is a convex subset of V then (2.2.6)
^K{X) = \ ' 1^ Hoo,
.^ , ^ It X f K
is convex, and cpx is lower semicontinuous if and only if K is closed. We have (pK = H where (2.2.7)
if(0 = sup(x,0, xeK
i^V,
is called the supporting function of K. That (fK just takes the values 0 and +00 is equivalent to t^K = ^K for every t > 0, and therefore equivalent to positive homogeneity of H, HitO = mo,
t>0,^£V'.
Hence the following theorem is a consequence of Theorem 2.2.4:
70
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
T h e o r e m 2.2.8. If K C V is closed and convex, then the supporting function H defined by (2.2.7) is lower semicontinuous, convex and positively homogeneous, that is, (2.2.8) ^ ( A i ^ + A26) < AiF(^i) + A 2 F ( 6 ) , if Ai > 0, A2 > 0, ^ 1 , 6 e r . Conversely, every such function H in V is the supporting function of one and only one closed convex set K CV, and it is defined by (2.2.9)
K = {xe
V; {x, 0 < H{0,
V^ G V'}.
Note that UK = V then H{0) = 0 but i f ( ^ = +00 when
C^O.
T h e o r e m 2.2.9. Let Ki,...,K^ be closed convex subsets of V with supporting functions i i f i , . . . , H^. If K = Ki H • " H Kjsf ^ 0 then the supporting function of K is the lower semicontinuous regularization of N
If K = ^ then hm^_Q h{^) =  0 0 . If Ki is compact, then either h = —00 or else h is the supporting function of a convex compact set. Proof. The first statement follows from Theorem 2.2.5. In any case h is positively homogeneous. If lim^_^Q /i( —oc then h is bounded from below in a neighborhood of the origin, hence > — 00 everywhere, so the lower semicontinuous regularization of h is the supporting function of a set k C K, hence K ^ 0. Since HjOj + 6>) < Hj{^j) + Hj{e) we have If Ki is compact then Hi is continuous, and it follows that h is continuous or = —00, which proves the last statement. Remark. Theorem 2.2.9 gives another proof of Helly's theorem (Theorem 2.1.6) for closed sets one of which is compact. In fact, if h{0) < 0 then Caratheodory's theorem applied to the epigraphs of Hj (see the proof of Theorem 2.1.3') proves that this remains true if we just keep n + 1 of the sets Kj suitably chosen. E x e r c i s e 2.2.2. With the notation in Theorem 2.2.9, describe a set with the supporting function if 1 H h H^ and a set with the supporting function maxj=:i^...^jv HjLet K now be a closed convex set containing the origin, and denote the supporting function of K by H. That 0 G if is equivalent to if > 0. Set (2.2.10)
K° = U; ^ G V, {x, 0 < 1 Va; G i^} = U ; ^ G V, H{0
< 1}.
It is clear that K° is closed, convex and contains the origin. One calls K° the polar of K.
THE LEGENDRE TRANSFORMATION
71
T h e o r e m 2.2.10. The polar K"" defined by (2.2.10) of a closed convex set K CV containing the origin is a closed convex set in V containing the origin, and we have (2.2.11)
K = {x; xeV,
{x, ^ < 1 V^ G X ° } .
Proof. It is obvious that {K°y D K. If x G (i^°)°, then ( x , 0 < 1 when ^ G K"", that is, when i J ( 0 < 1, and it follows that {x,^) < H{0 for all (. This is obvious if H{(,) = oo and otherwise we obtain {x^t^) < 1 when tH{() = H{t^) < 1, t > 0, and that means precisely that {x,(,) < H{(,). Hence x G if by (2.2.9), which proves (2.2.11). The other parts of the statement have already been verified. Exercise 2.2.3. Show that if K is a closed convex cone then K° is a closed convex cone, if°U;^eF',(x,0.) If iiT is a compact convex set with the origin as interior point then H{() < cxD for every (^, and H is the distance function associated with the polar K° of K, which is also convex and compact with the origin as interior point. Hence the supporting function iif° of K° is the distance function associated with K^ and we have (2.2.12)
H{0
= snp{x,0/H''{x);
H%x) =
sup{x,0/H{0
If K is symmetric with respect to the origin, then K° is also symmetric, and (2.2.12) is the wellknown relation between dual norms. Exercise 2.2.4. Let (p be an even nonconstant convex function on R with (p{0) = 0, and denote the Legendre transform by ip. Show that n
if = {:Z;GR^; J](^(xj) < 1} 1
is convex and closed, and that n
2_]xjyj 1
n
< max(l, y^ip{xj)),
if x G R'^, y G K°.
1
Deduce that X^^ ^(T/^) < 1 if 2/ G if °. From Theorem 2.1.22 we know that a convex function is Gateau differentiable in the relative interior of the set where it is finite. The differential is convex and positively homogeneous, so it is the supporting function of a set in the dual space. It has a natural interpretation in terms of the Legendre transform:
72
11. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
T h e o r e m 2.2.11. Let f be a convex lower semicontinuous function in V which is Rnite in a neighborhood ofxeV. Then the Gateau differential y j> f'{x; y) at x is the supporting function of K = {^;^eV',{x,0
= fix) +
f{0}
Proof. If ^ is in the subset Ki of V with supporting function f^{x; •), then f{x + y)f{x)>r{x;y)>{y,0. V^V, and we obtain 0 > sup((y, 0  fi^ + 2/) + fix)) yev
= fix)  {x, 0 +
no
Since f{x) + / ( ^ ) > (x,^) for 3\\ x e V and ^ E V, it follows that f{x) + / ( O — (^5^5 SO ( e K. On the other hand, if ^ G ii' we have by definition fix + y) > {x + y,0 so f'{x\y) claimed.
 ho
= (y,0 + / W ,
y^V,
> (y," "> 2/71)5 which are arbitrary points in the simplices X and y , m
X = {x 6 R'";a;i > 0 , . . . , a ; ^ > 0 , ^ 0 ; ^ = 1}, (2.2.13)
/ y  { j / G R " ; ? / i > 0 , . . . , y „ > 0 , 5 3 y f c = l}.
Then A expects to pay the amount m
(2.2.14)
A{x,y)
n
= ^^ay^x^j/fc.
THE LEGENDRE TRANSFORMATION
73
For a given strategy x oi A the player B can choose his strategy y so that A must pay raaxy^y A{x^y). If A chooses x to minimize his cost, he can count on paying at most
no matter how B bets. On the other hand B can choose his strategy so that A must pay at least\ maxmin A(x,y) yeY xex no matter how he bets. The following theorem shows that these two values are equal, so two rational players may as well pay this amount and quit! T h e o r e m 2.2.12. Let A be the real bilinear form (2.2.14), and define X,Y by (2.2.13). Then (2.2.15)
maxminA(x,2/) = minmax A(x,?/). yeY xex xex yeY
If s is this common value one can therefore choose x^ £ X, y^ E Y such that A(x^7/) <s, ye y ; A(a;,7/^) >s, xeX. Proof. The righthand side of (2.2.15) is min/i(x), xex
fi{x) = maxA{x,y) yeY
= max (:r,afc), ajt == (aijt, • • • ,ttmfc)l 0, {x, ak) > Ck, k = 1,..., L = {yeR'^;y>0,
,a^ €
{y,a;) < bj, j =
m},
l,...,n};
here x > 0 and y > 0 means that all coordinates are nonnegative, a* e R  is defined by YJ^{^.cik)yk  E i (2/,^*)^^'
and
Proof. The closed convex set K is not empty since ( t , . . . , t) G K for large positive t. Since inf {x,b) = — sup (a;, —b) = —HK{—b), the problem is to give another expression for the Legendre transform of the function / which is 0 in iiT and +00 in CK. Let g{x) = 0 when 0 < X G R^ and let g{x) = +00 otherwise, and let hk{x) = 0 when x G R'^ and {x,ak) > Ck^ hk{x) = +00 otherwise. Then g{(,) = 0 when ^ < 0 and g{(,) = +00 otherwise (cf. Exercise 2.2.3), and hk{0 — ^^k if ^ = ^Cbk^ A < 0, and hk{() = +00 otherwise. Since f = g {^hk it follows from Theorem 2.2.5 that / ( ^ ) is the lower semicontinuous regularization of m
m
F{0 = i n f i j ; XkCk] A  ( A i , . . . , A^) < 0, ] ^ A^a^ > 0 If all coordinates ^j are negative, then F{^) < 00, so F is then continuous, hence equal to / . In particular, replacing A by —y we obtain m
f{b)
= F{b)
^ snp{Y^ykCk]y
m
> 0 , ^ ^ ^ ^ ^ < b}.
GEOMETRIC INEQUALITIES
75
The condition YlVk^^k < & can be written m
{b,x) > ^yk{x,ak) 1
n
= ^Xj{y,a)),
x > 0,
1
so it means that {y^a^) < bj, j = 1,... ^n^ which completes the proof. The advantage of results like (2.2.16) is that they allow one to find both upper and lower bounds for the quantity in question. 2.3. G e o m e t r i c inequalities. Let F be a vector space of finite dimension n, and denote by IC{V) or /C for short the set of convex compact subsets of V. In /C we have a natural partial vector space structure: If Ki,K2 ^ IC and Ai, A2 > 0, we can define (2.3.1)
A1K14A2K2 = {AiXiHA2X2;xi eKi,X2
e K2}.
In terms of the corresponding supporting functions this is equivalent to (2.3.1)'
Hx^KihX2K2 =
^IHKI
+
^2HK2'
Thus the map K \^ HK from /C to the vector space Ci{V') of continuous functions on the dual space F ' , which are homogeneous of degree 1, identifies /C with a convex cone in Ci{V'). The identification carries the operation defined by (2.3.1) to the standard operation on functions, so we conclude that all the usual rules of computation are valid for (2.3.1). We shall write W = {HK\K G /C}. The linear space W — W is much smaller than Ci{V'). We can regard /C as a convex cone in this vector space. Fix a Lebesgue measure in F , for example, by choosing a basis for F or a Euclidean metric. Then the volume Q3(if) is well defined for every K e IC. (We could avoid fixing a measure by regarding 2T(if) as an element in the space of translation invariant densities on V.) The following basic theorem is due to Minkowski (see Appendix A for the definition of a polynomial): T h e o r e m 2.3.1. The map IC 3 K \^ V{K) mial of degree d i m F .
is a homogeneous
polyno
In the proof of the theorem as well as later results in this section it is convenient to approximate general elements in /C either by polyhedra or smoothly bounded ones: L e m m a 2.3.2. If K e IC and ft is a neighborhood find Ki^K2 E )C such that K cKiCn,
K
cK2Cn,
of K, then one can
76
IL CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Ki is a convex polyhedron with interior points, and K2 has C^ with strictly positive Gaussian curvature.
boundary
Proof. We may assume that J7 is convex. Every point in K is in the interior of some simplex with vertices in Q. We can cover K by finitely many such simplices. The convex hull of all of them has the properties required of Ki. When constructing K2 we may assume that the origin is an interior point oi K. Denote the corresponding distance function by p{x)^ so that K = {xe V]p{x) < 1}, and let 0 < x ^ C^{V)^ Jxdx = l. Then the C ^ function Pe{x) =
p{x
ey)x{y) dy + £\xf,
5 > 0,
is strictly convex, since p is convex, and for small e Pe{x) < p{x) h Ci£ < 1 + CiS, X eK]
ps{x) > C2 > 1, X
^n.
For small e the unique minimum point of p^ belongs to K, so K2 = {x;p^{x) < 1 + 2Cie} has C^ boundary with strictly positive principal curvatures since pe is strictly convex, and if C if2 C Jl if 2Cie < C2 — 1. The proof is complete. Proof of Theorem 2.3.1. Let i f i , K2 be polyhedra with the origin as interior point, and denote the supporting functions by ii"i and H2. If Ai, A2 > 0 then the supporting function of Aiifi + A2if2 is ^iHi + X2H2. I f ^ G F ' \ { 0 } , then (2.3.2) {x e Aiifi + A2if2; {x^O = A i i i i ( 0 + X2H2{0} = Aiifi(e) 4 X2K2{0.
K,{0 =
{^eKj;{x,0=Hj{0]
Let ^^, u — 1 , . . . , A/", be the directions such that the set on the left has dimension n — 1; it is clear that this condition is independent of the choice of the positive numbers Ai, A2. By Theorem 2.1.21 the boundary of Aiifi + A2if2 is the union of the sets (2.3.2) with i — iv, and these can only have sets of dimension n — 2 in common. Hence N
5J(Aiifi + A2if2)  5^2J(ch(0, Aiifi(^,) + A2if2(^.))) 1 ^ =  5 ] ( A i i f i ( ^ , ) + A 2 i i 2 ( e . ) m . ( A i i f i ( e . ) + A2if2(^.)). ^
1
GEOMETRIC INEQUALITIES
77
Here 03^ denotes the measure ^ ( (  , 0 ) ^^ ^he hyperplane {x G V] (x,^) = 0}, transported to the parallel supporting plane. We may assume that Theorem 2.3.1 has already been proved for lower dimensions, for it is obvious when n = 1. Then we know that 53^(AiKi(^) + X2K2{0) is a polynomial in Ai,A2 > 0, of degree n — 1, and the theorem follows for polyhedra in dimension n. In the general case we can use Lemma 2.3.2 to choose sequences of polyhedra K^ IKj, /i^oo. Then ^{XiKi
+ X2K2) = lim 2J(Aiiff +
X2K^),
fJL—KX)
and since the limit of a polynomial of degree n is a polynomial of degree n, the theorem follows in general. As shown in the appendix it follows from Theorem 2.3.1 that we can polarize the volume function: D e f i n i t i o n 2.3.3. The symmetric nlinear form 93(iCi,..., Kn) on IC^ with Q3(if) = 53(jFf,... ,i oo of the same expression with f± replaced by the defining functions / ^ of K^. Here ip is any continuous function on the unit sphere. But this follows at once since / ^ ^ f± uniformly in A;, and Theorem 2.1.22 shows that for x' E k minus a null set the functions f± and / ^ are all diff"erentiable and df*^{x') —> df±{x') as /x ^ oo. (Note that we have a uniform bound for these derivatives.) Thus we have defined the surface measure dS and defined the exterior unit normal almost everywhere with respect to dS. In view of Lemma 2.3.2 we have at the same time proved: P r o p o s i t i o n 2.3.4. IfV points, then (2.3.4) is vaUd.
is Euclidean, B,K
e IC and K has interior
Remark. With V Euclidean and B equal to the unit ball we have in particular that n^{B^K^... ^K) is the Euclidean area of dK. Also for more general, nonconvex sets K, one can adopt the definition ]h^{^{K
+ eB) 
V{K))/e
as the definition of the area (outer Minkowski area) of dK. Prom the proof given above it is easy to see that this agrees with the usual definition whenever dK is smooth. Taking for B some other convex compact set with interior points we get a definition of area with respect to a general distance function as in Theorem 2.1.23. Another interesting case of (2.3.4) occurs when we take for B an interval By = ch{0,y} or equivalently ch{ —^,  ^ } , where y EV. Then
nQJ(B„if,...,i^) =  /
\{y,a^))\dS{x)
JdK
is equal to the Euclidean length of y times the area of the projection of K in the plane perpendicular to y. Note that y »—> ^{By, K,... ,K) is convex,
80
IL CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
positively homogeneous and symmetric, so a symmetric convex compact set in V and case where K is the unit ball, n^{By, K,... (n — l)dimensional unit ball if \y\ — 1, so Cni = \ l
it is the supporting function of a norm in V. In the particular ,K) is the volume Cni of the we have
\{y.i)\du^{i).
M1,
where du is the surface measure on the unit sphere. Hence we obtain for any K E IC nf
V{By,K,...,K)du{y)
= \ j
J\y\ = l
dS{x) [
JdK
(y,^(a;)) da;(y)
J\y\ = l
= Cn1 [
dS{x).
JdK
This means that Cni times the area of dK is the integral over y E S"^'^ of the area of the projection in the orthogonal plane of y. We shall now return to the mixed volume and prove another basic result, essentially due to Minkowski: T h e o r e m 2.3.5. IfK', Kj G /C and Kj C K'j for j = 1 , . . . , n, then (2.3.6)
0 < 2 J ( i ^ i , . . . , Kn) < « ( i ^ I , . . •,
K).
Proof. The statement is obvious when n = 1, so we assume that n > 1 and that the theorem has already been proved in dimension n — 1. It suffices to prove that ^(Ki^... ^Kni^Kn) < 9 J ( i f i , . . . ,iir^_i,if^), for iteration of this result gives the second inequality (2.3.6) in view of the symmetry, and if Kj consists of a single point in Kj, then it follows that ^{Ki,..., Kn) > » ( i f { ' , . . . , i^n) = 0 In view of Lemma 2.3.2 we may assume in the proof that Ki,..., Kni are polyhedra with interior points, and we assume that V is Euclidean. For any polyhedron K with interior points it follows from (2.3.4) that N
(2.3.7)
n 2 J ( K , . . . , K, K J  5 ] F x ^
{^uMK{^,))
where d denotes (n — l)dimensional volume in a hyperplane, K{0
=
{X£K;{X,0=HK{0},
GEOMETRIC INEQUALITIES
81
and the sum is taken over the exterior unit normals of all the (n — 1)dimensional faces of K. li K = XiKi + • • • + X^iKni and A i , . . . , A^i are positive, then (2.3.2)'
K{0
= XiKiiO
+ ••• +
XniKndO
The vectors ^^ in (2.3.7) are then those for which this is a set of dimension n — 1, so they do not depend on A. The sets Kj{^) lie in parallel planes, but we can shift them by parallel translation to the plane [x] (x^^) = 0}, so D ( A I K I ( ^ ) H h XniKjiiiO) is equal to ( n  l ) ! A i  .  A n _ i t ) ( i f i ( 0 , . . . , i f n  i ( 0 ) + where the dots indicate terms not divisible by all Aj, j = 1 , . . . , n — 1, and the (n — l)dimensional volumes are defined using the parallel translation just mentioned. (Recall that (mixed) volumes are invariant for separate translation of each argument.) Hence
V
By the inductive hypothesis the coefiicient oiHKy^i^iv) here is nonnegative. If Kn C K'^ then HK^ < HK^, SO it follows that ^{Ki,.. .,Kni,Kn) < QJ(i^i,..., Kni>K'^), which completes the proof. We shall now make the left inequality (2.3.6) more precise; for the sake of convenience only we assume that 0 belongs to all the bodies: T h e o r e m 2.3.6. If Ki,... conditions are equivalent:
,Kn e IC and 0 G n'^Kj,
then the following
(i) ViKu...,K^)>0. (ii) One can choose Cj G Kj such that e i , . . . , e„ is a basis for V. (iii) Ifl n'^{R 
rf
GEOMETRIC INEQUALITIES
83
since the roots are real and differ at least by R — r. This is a strong form of the isoperimetric inequality. Thus the polynomial ^{K) is strictly hyperbolic in the direction 5 , considered in the vector space generated by convex bodies with nonempty interior, and then it is strictly hyperbolic in all such directions. This will prove Theorem 2.3.7 when n = 2. Proof of Theorem 2.3.8. (See Burago and Zalgaller [1, Section 1.3].) It is sufficient to prove the theorem when if is a convex polygon and r < X < R. Consider a circle of radius A with center which slides along dK. When it slides along a side we see that each half circle bounded by points with parallel tangent will slide over a set of area 2AL where L is the length of the side (Cavalieri's principle). Counted with multiplicities the area covered by the circle when the center travels once around dK is equal to ^XL{dK). The multiplicity can only be in doubt at points which have distance exactly A to a vertex or a side, and these are of measure 0. Otherwise the multiplicity with which a point x is covered is decided by counting the number of times the circle with center at x and radius A intersects dL. The multiplicity must always be even, for a circle starting outside K will finish outside K. If the multiplicity is 0 at a point x^ then the circle of radius A with center X is either entirely in the interior of K or entirely in the exterior, which cannot happen if r < A < i? and the distance from x to if is < A. Thus K f \B is entirely covered at least twice, apart from a null set, so we have 5J(if + \B) < (4AL(aif))  4AQJ(if, 5 ) , which proves the theorem. Proof of Theorem 2.3.7. Throughout the proof we shall assume that n > 2, and we shall no longer work with convex polyhedra but shift to convex sets with C^ boundary of positive Gaussian curvature with the origin as interior point. We shall denote this subset of /C by /Creg When K G /Creg we can rewrite (2.3.4) by introducing ^{x) E S''^"^ as a new variable. Recall that dK 3 x f> ^(x) G S'^~^ is a diffeomorphism. We extend the inverse gni 3 ^ )_^ x(^) G dK to R'^ \ 0 as a positively homogeneous function of degree 0. Then H{(^) = (x(^),(^), and since (dx,^) = 0 on Ta,(^)5M, it follows that H'{(^) = x{^). The surface area duj{^) on S'^~^ is equal to hidS{x) where K is the Gaussian curvature at x G dK. The principal curvatures and the radii of curvature are defined by the equations n
dxj{i) = Yl ^jkiOd^k == Rd^j, i = 1,...,n, k=i
n
Yijd^^j = 0. 1
Here HjkiO — d'^HiOl^^jd^kSince H'{l^) is homogeneous of degree 0, the radial direction is an eigenvector with eigenvalue 0 while the other
84
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
eigenvalues corresponding to the principal curvatures are solutions of the equation R^ det{Hjk  R6jk) = 0. The leading term is (—i?)"^"^, so the product K~^ of the roots is the constant term, that is, with Id denoting the unit matrix, «  '  ^ d e t ( i ? " ( 0 + i?Id)fl=o. Denote by D{A) the determinant of a symmetric nxn matrix A^ and let D be the polarization of D. (We recall from the example after Corollary 2.1.32 that D is hyperbolic with respect to the cone of positive definite matrices.) Thus we obtain
^'^nD{H"{0,...,H"{OM), and we can rewrite (2.3.4) when B ^ K and K G /Creg in the form
By polarization of this polynomial in K we obtain for if i , . . . , Kn G /Creg (2.3.4)' Q J ( K i , K 2 , . . . , i f n ) = /
H,{OD{H!^{0,.H':{OM)du;{0.
where Hj = HKJ is the supporting function of Kj. (Note that (2.3.4)' gives another proof of Theorem 2.3.5 since D{H2^... ,H!^,ld) > 0 by Corollary 2.1.32.) The lefthand side is symmetric in Ki,... ^Kn^ so this must also be the case for the righthand side. Thus we may exchange the index 1 with any one of the indices 2 , . . . , n in the righthand side. Our goal is to prove (2.3.8), which means that for the extension of the mixed volume function to (/C — /C)^ (2.3.8)'
2J(fc,fc,i f 3 , . . . , ifn) < 0,
if Q3(fc, K 2 , . . . , i^n) = 0,
when k is in the plane spanned by Ki and K2. (Note that if 93(fc, 7^2? • • • 5 Kn) is equal to 0 then Q3(AiA; + X2K2, Xik + AziiTz, i^s, • • •) = X'Mk,
A:, i f a , . . . ) + ^MK2,
i^s, i^3, • • •)
is not positive definite if and only if the coefficient of Af is < 0.) That k is in the plane spanned by Ki and K2 is no significant restriction on
GEOMETRIC INEQUALITIES
85
the smooth homogeneous "supporting function" Z of A:, for Z + IH^ will for large positive i be strictly convex nonradially, hence the supporting function of some K\ E /Creg Rewriting (2.3.8)' using (2.3.4)', we conclude that what must be proved is that (2.3.10)
/
Z{i)D{Z"{i\E'i{i\...,K{i)M)i^{i) 1. Now the side condition in (2.3.10) means precisely that Z is orthogonal to the harmonics of degree 0, the constants, so we conclude that (2.3.10)
86
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
is valid for iir2 = * • • = Hn = H with strict inequality except when Z is linear. (The vanishing for linear Z is no surprise for it expresses precisely the translation invariance of the mixed volumes.) The extension of (2.3.10) to general K2,... ^K^ G /Creg will now be carried out by a continuity argument, adapted by Alexandrov from Hilbert's proof of the BrunnMinkowski inequality when n == 3. The symmetric bilinear form corresponding to the quadratic form in (2.3.10) is (W, Z)^
j
W{OD{Z"{il
H'^{i\
. . . , H';[i\
Id) du;{i\
for the symmetry of the righthand side inW^Ze /Creg extends by linearity to W^Z e /Creg — /Creg The differential operator
z^D{z"{0,HUO,,K(OM) is elliptic, with negative principal symbol. To verify this we make it explicit at (^ = ( 1 , 0 , . . . , 0) again. As above the coefficient
n\D{Z^'{0:H!;{0.',K{OM) of Ai •.. A, in D{XiZ'\0 + ^2H^iO + ' ' • + KiK(0 + A, Id) is equal to the coefficient of Ai • • • A^i in the determinant of AiZ''(^) + X2H'^{^) f • • • +• XniH!ll{^) with the first row and column removed. The principal symbol at the (co)tangent vector 0 = {0i,..., On) with ^i = 0 is obtained when Z"{^) is replaced by —6 (8) ^, so it is equal to
(2.3.13)
e^
Did®e,H'^io,...,Kio,id).
Apart from a factor 1/n the righthand side is not changed if we remove the first row and column. Since D is hyperbolic with respect to the cone of positive definite matrices and complete, and since 6 0 will be chosen in a moment. The eigenfunctions Z and eigenvalues A are solutions of the elliptic equation (2.3.15)
D{Z",H'^,...,H':,ld)=^XQZ.
GEOMETRIC INEQUALITIES
87
The side condition in (2.3.10) can be written
0=/ = /
zb{H'{,...,H';M)MO H2D{Z",Hl...,H';,ld)dw{i).
We choose (2.3.16)
Q = b{Hl...,H':,M)lH2
to make H2 a solution of (2.3.15) with A = 1. Note that both numerator and denominator are strictly positive. Then the side condition in (2.3.10) becomes orthogonality in the sense of (2.3.14) to this eigenfunction. Thus proving (2.3.10) means to show that all other eigenvalues are < 0. The next step is to determine the eigenspace with eigenvalue 0, that is, the solutions of the equation (2.3.15)'
b{Z",Hl...,H';M)
= Q.
By the ellipticity all eigenfunctions are in C^. In view of the hyperbolicity oiD and Corollary 2.1.32 it follows from (2.3.15)' that
b{z"{i), z"{0, H'l{ii..., K{OM) < 0, with strict inequality except when Z"{^) = 0. Since
0=/ = f
zb{z",Hi...,H';M)duj{o HsD{Z'\ Z\ R'l..., i/;', Id) rfa;(0
and jEfa > 0, we conclude that Z" = 0 so Z is a linear function. Conversely, every linear function satisfies (2.3.15)'. Thus the multiplicity of the eigenvalue zero is always equal to n, and the eigenfunctions are always the same. Now we deform our eigenvalue problem to the case of the ball studied above by introducing mj
= fiHj\{lfi)H,
j =  2 , . . . , n , 0 < / x < 1.
Here H is the supporting function of the unit ball. When /i = 0 we have only the simple eigenvalue 1, the eigenvalue 0 of multiplicity n, and otherwise negative eigenvalues. For any /x G [0,1] we know that 1 remains an
88
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
eigenvalue, with eigenfunction ^H2, and that 0 remains an eigenvalue with the same eigenspace. Hence standard ellipticity theory shows that no new eigenvalues > 0 can appear and that 1 remains a simple eigenvalue, which proves (2.3.10). In fact, for large enough M the inverse of the operator
z^Mz
^Q"^b{z",^H'^,... ,^^;',id) v ^
is compact in L^, uniformly continuous with respect to /x, has only a simple eigenvalue ( M — 1)~^ in (M~^, oo) when // = 0, and has M"^ as an eigenvalue of fixed multiplicity when /z G [0,1]. The eigenvector corresponding to the eigenvalue (M  1)~^ depends continuously on /x G [0,1]. By continuity it follows that the largest eigenvalue in the orthogonal space of this eigenvector and the eigenvectors with eigenvalue M~^ must remain < M~^ as is known when /x = 0. Remark. By Theorem 2.3.7 and Proposition A.2 the mixed volume has much of the properties we established in Section 2.1 for hyperbolic polynomials. It is therefore natural to ask if it might actually be hyperbolic. The answer is no for n > 2 by an example already given by Minkowski. Let K be the convex hull of a ball of radius R and a point XQ outside the ball, and let B be the unit ball. Then it is geometrically clear from a picture that K 4 hB only differs from (1 + h/R)K at a distance 0{h) from XQ. Hence V{K + hB) = (1 + hlRYV{K) 4 0(/i^), and since the coefficient of /i^ is QJ(5), it follows that V{K + hB)  (1 4 h/R)''V{K)
+ /i^(5J(B)  Q3(ii:)/i?^).
If we choose K so that V{K) — ^(B), then i? < 1, and any number R G (0,1) can occur. Thus the equation 5J(iir + hB) = 0 has the form (l + / i / i ? r = / i ^ ( i ?  ^  l ) which implies 1 + h/R = LJ{R''  1)^/^/^ where CJJ is an n t h root of unity. Hence there are at most two real roots, so there is a nonreal root if n > 2. A simple consequence of the FenchelAlexandrov inequalities is the BrunnMinkowski inequality (2.3.17)
^{Ki^K2)^
>QJ(iCi)n+9J(i^2)*,
i^i,K2e/C,
expressing the concavity of K H^ QJ(i^) ^, and its corollary (2.3.18)
V{Ku . . . , K u K 2 ) > Q 3 ( K i ) ' ^ 0 3 ( ^ 2 ) ^ .
(See Propositions A.l and A.2.) When K2 is the unit ball, then (2.3.18) states that the Euclidean area nV{Ki,..., K i , ^^2) of dKi divided by the
GEOMETRIC INEQUALITIES
89
volume of Ki raised to the power (n — l ) / n is at least as large as the value for the unit ball, since there is equality in (2.3.18) when Ki = K2. This is the isoperimetric inequality which has a very old history. Also (2.3.17) has been known since the turn of the century for convex sets. Since the 1930's it is also known for nonconvex sets. There is such an elementary proof that it cannot be resisted although it falls outside the main topic. We use the notation m for the measure rather than 93 to indicate that in this generality Lebesgue measure must be used. T h e o r e m 2.3.9. If A and B are compact subsets ofR^, {x {y]X ^ A^y E B} then (2.3.17)'
and A\ B =
m{A + 5 ) ^ / ^ > m ( ^ ) ^ / ^ f m ( 5 ) i / ^ .
Proof. It suffices to prove this when A and B are unions of finitely many disjoint (products of) intervals, and that can be done by induction over a\b if a and b are the number of intervals which constitute A and B. In fact, if A and B are both intervals, with side lengths a^, 6^, 2 = 1 , . . . , n, then
by the inequality between geometric and arithmetic means, if aj \ bj = 1 for every j . For homogeneity reasons this gives (2.3.17)' in general if A and B are both intervals. Now assume that a > 1 and that (3.2.17)' is proved already for smaller values of a H 6. Then there is a plane Xj — constant separating two of the intervals defining A, In view of the translation invariance it is no restriction to assume that it is the plane Xj = 0 , and that m(A+)/m(A_) = m(B+)/m(5_), if A± and B± are the intersections of A and B with the half spaces defined by Xj > 0 and Xj < 0 respectively. These are constituted by at most a — 1 and at most b intervals, so by the inductive hypothesis m{A + S ) > m{A^ + 5 + ) + m(A_ h BJ) > (m(A+)^/^ + m ( 5 + ) ^ / ^ ) ^ + (m(A_)^/^ + m ( S _ ) i / ^ ) ^ (m(A)^/^+m(5)^/^)^, for iim{A^) = Xm{A) then m(A_) = (1  A)m(A), m{B^) = Xm[B) and m{B) = (1  X)m{B). This completes the proof.
90
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
Remark. It is known that there is strict inequahty in (2.3.17)' unless m{A + 5 ) = 0, or A or B consists of a single point, or A and B are convex sets which differ only by a translation and a homothety. For a proof we refer to Burago and Zalgaller [1, §8.2]. Chapter 4 of the same book contains an extensive discussion of applications of the FenchelAlexandrov inequalities and also an algebraic proof. Prom Theorem 2.3.7 and Proposition A.2 one can obtain numerous inequalities between the mixed volumes, but we shall not write them down here. Instead we shall end our geometric discussion by deriving some formulas for mixed volumes with the unit ball B in a Euclidean space. We shall write h{^) = \^\ for the supporting function of B. Let K G /Creg have supporting function iJ, and let (2.3.19)
W^ = 5 J ( K , . . . , i^, S , . . . , B).
If 0 < z/ < n it follows from (2.3.4)' that
JS^^
'
V
' ^
V
'
Assuming as in similar calculations above that ^ == ( 1 , 0 , . . . , 0) we find as there that the coefficient of Ai • • • A^ in D{{\i
+ • • • + X.)H'\0
f (A,+i + .. • + \ni)h"{0
+ A, Id)
is equal to the coefficient of Ai • • • A^i in the determinant of aH"{C> + &Id,
a = Ai + • • • + A^, h = K^i
+ • • • + A^i,
with the first row and column removed. We may assume that H"{£^) is diagonal, with the principal radii of curvature i ? i , . . . , i i ^ _ i as diagonal elements. The determinant is then equal to Y\!x~ (O'Rj^b)^ so the coefficient of a^6^~^~^ is the z/th elementary symmetric function { i ? i , . . . , Rj^}. Thus the coefficient of Ai • • • A^_i is equal to { i ? i , . . . , R^,} times u\{n — u — 1)!, and after dividing by n! we obtain (2.3.20)
^ n  v ^ — i ^ l {Ri,...,R^}du, n( ^ j J 5  1
v
0, then
92
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
is for small r > 0 the volume of the set of points at distance < r from dK. Following Weyl [1] we shall now study this volume not only for convex hypersurfaces in R'^ but for arbitrary smooth compact submanifolds M of a Euclidean space. It is convenient to change notation now, so we shall denote the dimension of M by n and assume M embedded in R*^"^^ for some p > I. Let N{M) be the normal bundle of M . For sufficiently small r > 0 the map N{M) 3 (x,n) ny x + n G R^"^^ is a diffeomorphism of {(x,n) G iV(M); n < r} on the set of points in R ' ^ + P at distance < r from M. To calculate the volume we parametrize a piece of M by uj 3 X i^ f{x) e M , where cu is an open subset of R"^. We assume the coordinate patch u chosen so small that there is an orthonormal basis ni(x)^... ^Xip{x) for Nf(^x^(M) depending smoothly on x. The considered map is then p
LjxRP
3 (x, t) ^ F{x, t) = f{x) + Y^
t^nj,{x).
1
We have dF/dtj^ l,...,n
— n^, and modulo the normal plane we have for j = p
p
dF/dxj = fj + j^Udjn,
n
= /i  E
1^=1
E
t(hij,n,)g'''fk
u=li,k=l
where hij G N are the coefficients of the second fundamental form and fj^ = dkf (See e.g. Klingenberg [1] for the Riemannian geometry used here and below.) The volume spanned by the partial derivatives of F is therefore equal to the volume y/g spanned by / i , . . . , / n multiplied by the determinant of {6jk  YZ=i^^(^h^''^')lk=i^ where h^ = Y^i^i ^ijQ^''Hence the volume of the tube over f{uj) is
(2.3.21)
/ ,/^dx I JLJ
det((5,,
J\t\0 (2.3.22)
/
(t,a)^dt = r^+^arCp
T]
^ ^ '
a G R^,
where Cp is the volume of the unit ball in R^ which makes (2.3.22) obvious when a = 0. For reasons of homogeneity and rotational invariance it suffices
GEOMETRIC INEQUALITIES
93
to prove (2.3.22) when a = ( 1 , 0 , . . . ,0) and r = 1. Then the integral is equal to
= Cp_iS((a+i), i(p+i)) = c,.ir((a+i))r((^+i))/r((a4p)+i). When a is replaced by a + 2 the righthand side is multiplied by the factor {a + l)/(c»" hp + 2), so (2.3.22) follows by induction. If we polarize the two sides it follows more generally that for a i , . . . , a^j G R^ (2.3.22)'
/
{t,ai){t,a,)dt
= r''+PECp
TT
 i  ,
where and the sum is extended over all a\ permutations i / i , . . . , z/^^ of 1 , . . . , a. (We have used that a\ = a\\{a  1)!!.) The integral in (2.3.22)' vanishes when (7 is odd. If we expand the determinant in (2.3.21) it follows that the inner integral of the terms of even degree 2a with respect to t is equal to
Here the summation runs over all atuples of unequal integers a and P chosen among the integers 1 , . . . , n which are permutations of each other with signature sgn (^). If we let /3i and P2 change places we can replace the first factor by
by the Gauss equations. Hence we obtain T h e o r e m 2.3.10. For a manifold M of dimension n embedded in R^+^, the volume of the set of points at distance < r from M is for small r > 0 equal to
(2.3.23)
c,
Y: 0 0 it is positive if X2 is small enough. Since d^f > 0 when X2 7^ 0 we conclude that there is a unique solution xs = X{x2) which is a C^ increasing function of \x2\ for small 0:2 7^ 0 and —> 0 as X2 ^^ 0. By (2.4.3) we have F"{x2) = dlf(x2,X)

{d2dsf{x2,X))ydlf{x2,X).
Hence ]i^F"{x2) 2v{0), solim^2_+oi^"(^2) > 2^7(0) also. On the other hand, id2d,f{x2,X))'/dlf{x2,X)
= ixlv'{Xy/{xlv"{X)
+
< with equality iiu"{X)
u"{X))
W{Xf/v"{X),
= 0. Hence
lim F"{x2) = 2(^(0)  2t;'(0)V^"(0)) iiu" has zeros arbitrarily close to the origin. Such functions exist, although they cannot be analytic, and for them we have lim F'\x2) X2+0
= 2(i;(0)  2^'(0)V^"(0)) < 2^(0) 
Ih^
F'\x2),
a:2>H0
which proves that F ^ C'^. It is easy to find a convex set with C^ which is defined by (2.4.4) near 0.
boundary
In the example the second derivative of F was bounded although not continuous. This suggests that one should consider the regularity condition in the following definition:
SMOOTHNESS OF CONVEX SETS
97
Definition 2.4.2. A closed set K C R^ is said to have C^'^ boundary dK if for every point XQ G dK one can choose coordinates with the origin at XQ and a neighborhood U of XQ such that (2.4.5)
UnK
= {xeKxi>
f{x')},
where x' — {x2^ •.. ,x^) and / E C^'^, that is, / G C^, and df is Lipschitz continuous. For convex sets this condition takes a very simple form: P r o p o s i t i o n 2.4.3. Let K G /C(R^). Then dK G C^'^ if and only if there exists some R> 0 such that K is the union of balls with radius R. Proof. Assume that (2.4.5) holds with U = {x;\x\ /'(O)  0 and \nx^)ny')\ f{x') if \x\'^/2R > C\x'\^{2^ which is true if CR < 1. This gives the desired ball containing the origin. To handle points {f{x'),x') near 0 we recall that \f'{x')\ < C\x'\ and that \f{x')\ < C\x'\^/2. There is a coordinate change differing from the identity by 0(:z;') which shifts the origin to this point and the direction (1, —f'{x')) to the direction of the new r^iaxis, so we get the same conclusion at this point with C replaced by C ( l + 0(:r')) < 2C if \x'\ is small enough. If R is sufficiently small it follows that there is a ball with radius R contained in K and with any given point in dK on its boundary. Every point x £ K can be written X = Xxi + (1 — X)x2 where 0 < A < 1 and Xj G dK. If Bj C if is a ball of radius R containing Xj, then XBi + (1 — A)52 is a ball with radius R contained in K and containing x. Now assume conversely that the interior ball condition in the proposition is fulfilled by K. Choose the origin at an arbitrary point in dK so that K is contained in the half plane where xi > 0. Then the ball in the hypothesis must be defined by xp < 2i?xi, so it follows that if fl C/ is defined by ^i > / ( ^ ' ) with / convex, iiU = {x; :z; < R}. Since 0 < f{x')
X2 sinxa}, then the projection ivK in the X1X2 plane is defined by xi > —\x2\ Thus the boundary is C^ except at the origin where it is only Lipschitz continuous. Exercise 2.4.1. Show that if iiTi, ^ 2 ^ /C(R'^) have boundaries in C^'^, then the boundary of Ki f K2 is also in C^^^. The regularity of projections and sums of convex sets has been studied in great detail by J. Boman and C. Kiselman. We refer to Kiselman [1] for a survey of the results. 2.5. P r o j e c t i v e convexity. If F is a vector space over R of dimension n + 1 , then the corresponding projective space P{V) is defined by identifying points in F \ {0} which are multiples of each other; thus P{V) is the space of onedimensional subspaces of V. One defines d i m P ( y ) = n. li W is a linear subspace of the vector space V, then P{W) is called a subspace of P{V). It is a projective line if it has dimension 1 and a projective hyperplane if the dimension is n — 1. The identity dim 1^1 + dim 1^2 = dim(I^i + 1^2) + dim(I^i fl W2) valid for linear subspaces of V implies that d i m P ( t ^ i ) + dimP(I^2) = dimP(T^i + W2) + dimP(T^i H 1^2). Here P{Wi f 1^2) is the smallest subspace of P{V) containing P ( W i ) and P(1^2)5 cind one should interpret P({0}) as the empty set with dimension — 1. For any coordinate system XQ,...,Xn in V we can take
PROJECTIVE CONVEXITY
99
x i / x o , . . . ^Xn/x{) as coordinates in P{V) outside the projective hyperplane defined by XQ = 0, so P{V) is identified with R"^ extended by a "plane at infinity", a projective space of dimension n — 1. The coordinates XQ, . . . , x^ are called homogeneous coordinates, and we shall often use the notation {XQ : xi : •' • : Xn) for a point in P^ = P(R'^"'"^). More generally, for any vector space W the projective space P ( R 0 W) can be identified with the union of W and the plane P{W) at infinity. If Vi —> V2 is an injective linear map between vector spaces, then the induced map P{Vi) —> ^ ( ^ 2 ) is called projective. In a projective space P{V) a projective hyperplane is the image of a hyperplane W CV, defined by an equation L == 0 where L is a linear form on V. We can then identify P{V) \ P{W) with {x G VL{x) = 1}. This is an affine space which becomes a vector space if we choose a point in it as origin. This affine structure is independent of the choice of L. Thus P{V) \ P{W) has an affine structure which makes the notion of convex subset in the usual affine sense meaningful. The projective line is topologically a circle. Two different points in it determine two intervals bounded by them. If we have four different points A = {ao : a i ) , B = {bo : 61), C = {CQ : ci), D = {do : di) in the projective line P ( R ^ ) , then the cross ratio is defined by rA r? ry Ti\
CLQCI OQCI
 aiCo /aodiaido  bico I bodi  bido
note that it is homogeneous of degree 0 in each variable in R^ and therefore defined on the projective line P^. lia^ = bo = CQ = do = 1 it is the quotient between the ratio in which C divides AB and that in which D divides AB. If we make a linear transformation in R^, then each factor is multiplied by the determinant and the cross ratio is unchanged. Hence it is invariantly defined on any projective line. Note that C and D lie in the same open interval bounded by A and B if and only if {A, B] C, D) > 0. Definition 2.5.1. A set K in a projective space P{V) is called projectively convex if every straight line in P{V) intersects K in an interval (which may be open, half open or closed, empty, a point, the whole line, the whole line except one point, or an interval with two end points). The definition is clearly symmetric under passage to the complement, that is, K is projectively convex if and only HCK = P{V)\K is projectively convex. We shall say that a pair of sets Ki^K2 C P{V) is convex if K2 = ZKI and K i , hence 7^2? is projectively convex. Both Ki and K2 are then connected. In terms of the cross ratio the definition can also be stated as follows: lixi^yi G Ki and X2^y2 ^ ^ 2 lie on the same line, then (^1,2/1; ^2,2/2) > 0.
100
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
If if is a convex subset of R"^ in the usual (affine) sense, then iiT is a projectively convex subset of P^;, for it has empty intersection with Hues contained in the plane at infinity and intersect other lines in an interval. Also the complement is projectively convex. Another example is obtained if we consider a quadratic form Q{x) = aoxl + aixl H
h a^a:^
and define K = {{XQ : • • • : Xn); Q(^) > 0}. Since a quadratic form in R^ is positive either in R^ \ 0 or in a symmetric set bounded by one or two lines, it is clear that K is projectively convex. If there are at least two coefficients of each sign in Q then every hyperplane intersects both K and the complement of K, so it is not possible to choose coordinates to make K convex in the affine sense. This example is important; we shall see that essentially there are no projectively convex sets beyond those we have listed now. L e m m a 2.5.2. If K is a projectively convex subset of P{V) which is not contained in any subspace, then the interior K° is not empty, and it is dense in K. Proof. We prove the lemma by induction with respect to the dimension n \ 1 oi V. It is trivial when n = 1 so we assume that n > 1. For an arbitrary X G iiT we can choose X i , . . . , X^ G K such that X , X i , . . . , X^ are independent (that is, corresponding elements in V are linearly independent). We can choose coordinates in V so that X is the origin (1 : 0 : • • • : 0) and Xj is the point at infinity on the j t h coordinate axis. By the inductive hypothesis the intersection Kj of K with the coordinate plane 11^ where Xj = 0 for every j 7^ 0 has points arbitrarily close to the origin which are interior with respect to 11^. Let Uj C Kj be an open set in 11^ close to the origin with all coordinates except Xj diff'erent from 0. If yj G /7j, j = 1, 2 then there is a unique line between them and one of the intervals with end points yi,7/2 isin K. If the whole lines are in K then K° D U1UU2. On the other hand, if there is a point z ^ K on the line determined by y 1,2/2 5 then all lines nearby through z cut both Ui and U2, and the interval determined by these intersections not containing z must be in K. Thus 2/1 and 2/2 are in the closure of iC°, and the proof is complete. If Ki^K2 (2.5.1)
is a convex pair in P{V), we shall write V = ~K[f\'K~2 = dKi = dK2,
thus P{V) = KlUTl}
K^.
This boundary never has any interior points. In fact, assume that U C T is open. Since U H Kj = 0, it follows from Lemma 2.5.2 that U H Kj = 0 unless Kj is contained in a hyperplane 11. Then C/ C F C 11, so C/ is empty.
PROJECTIVE CONVEXITY
101
If K is a projectively convex set in P{V) which is contained in a subspace P ( W ) , then K is projectively convex as a subset of P{W) and conversely, so we have a trivial reduction to lower dimensions unless K has interior points, by Lemma 2.5.2. From now on we shall therefore always assume that both Ki and K2 have interior points. Note that Ki^K2 and K°^K2 are also convex pairs then. In fact, if Ki 3 Xj —> x, Ki 3 yj ^ y then an interval bounded by Xj and yj is contained in iiTi, and after passage to a subsequence it converges to an interval bounded by x and y contained in Ki. Hence Ki is projectively convex, and so is K2. We shall primarily study convex pairs where one of the sets is open and the other closed, with interior points. L e m m a 2.5.3. If L is a line then we have either (i) L n r  0, and L C KI or L C K^; (ii) L n r is an interval, possibly reduced to a point, with complement in L contained either in K^ or in K2; (iii) L n r consists of two points and LHCT consists of one interval in KI and one in K^. Proof. Since Kj fl L is an interval, the intersection F fl L is empty or consists of one or two intervals, possibly reduced to points, and the complement in L consists of one or two open intervals / then. Since / is the union of the disjoint open sets / fl KI and / fl X  , the connectedness of / implies that one is empty. If there is only one interval I we have case (i) or case (ii). Now suppose that we have two such intervals, and that both are contained in K ° , say. Choose Xi,yi in the two intervals. They are separated by points ^2,2/2 ^ F n L. We can choose 2:2,2/2 ^ ^2 so close to X2,2/2 that the line between them intersects neighborhoods of Xi and yi contained in Ki, which contradicts the definition of convexity. This contradiction proves that we have one interval in if ° and one in K  . If a separating interval contains two points ^2,2/2 ^ F we get a contradiction in the same way, so F fl L must consist of precisely two points separating an interval C K^ and an interval C K2 The proof is complete. L e m m a 2.5.4. IfP{W) is a subspace ofP{V) such_that Tw = PjW)^^ has interior points relative to P{W), then P{W) C Ki or P{W) C ^ 2 , and Fv^^ is a projectively convex subset of P{W). Proof. First we prove that Fv^^ is the closure of its interior u with respect to P{W). By hypothesis a; is not empty. Ii x ^ Tw \uJ we consider the lines connecting a; to a point y £ UJ. If they are all contained in Vw^ then X is the vertex of an open cone C u, so the assertion is true. Assume now that on one such line there is a point z ^ Tw^ say z ^ K2, that is, z E K^. A line through x and a point ( G Ki fl P{W) close to z must still intersect
102
II. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
u) in an open nonempty set, so it follows from Lemma 2.5.3 (ii) that the interval between x and uj not containing (^ must be a subset of Vw Hence r^^ contains an open set in P(W) with x on its boundary. An immediate consequence is that Vw is projectively convex in P{W). In fact, if x,?/ G Tw^ then we can find Xj G u with Xj ^ x. By Lemma 2.5.3 one of the intervals bounded by Xj and y must be contained in Tw, for the intersection of the line through these points and Tw has a nonempty interior. Since Tw is closed it follows when j ^ oo that an interval bounded by x and y is contained in Fw If we join two points x,y e P{W) \ Tw by a line L, then the fact that L n r^^ is a closed interval implies that the complement, which contains x and y, is an open interval J. Hence we have one of the cases (i) or (ii) in Lemma 2.5.3, so x and y are both in K^ or both in K2, which proves that P{W) C K T or P{W) C ^ . The proof is complete. L e m m a 2.5.5. IfP{W) is a subspace ofP{V) such that P{W)nK] + 0, j = 1,2, then Fvi/ = P(VF) fl F has no mteiioi point, and the closure of
p(w) n K ; isP{W)nlK~j^ {P{w) n K ; ) U r ^ , J 1,2. Proof. That Tw has no interior point is a consequence of Lemma 2.5.4. We must prove that Tw is contained in the closure of Uj = P{W) 0 K^. It suffices to discuss the case j = 1. Let x G Vw and suppose that a neighborhood ?7 of x in P(W) does not meet uji. If the line L through x and a point y E uJi contains a point in a;2, then we have the case (iii) in Lemma 2.5.3 and uji contains an interval on L bounded by x since x G Tw This is a contradiction. Hence we must have the case (ii), so L C Ki, since y ^ K2, hence LnU C Tw But if this is true for all y ^ uJi, then Tw has interior points which is again a contradiction proving the lemma. L e m m a 2.5.6. IfxGT or L C 1^2
then there is a Une L with x E L and L C Ki
Proof. A point in T cannot be isolated, for then either Ki or K2 has an isolated point and must consist of that single point. Hence we can choose Xj ET\ {X} with Xj ^ X SiS j ^ 00. By Lemma 2.5.3 the line Lj through X and Xj is entirely contained in one of the sets Kk except in case (iii); in that case the whole line except the interval bounded by x and Xj close to x is contained in one K^. When j —> 00 we conclude that a limit of the lines Lj is contained in one of the sets K^. We can now study the twodimensional case. (In the onedimensional case a projectively convex set is by definition just an interval.) If Ki,K2 is a projectively convex pair in P^, then one of the closures, say K2, contains a line, which we can put at (X). Then K^ C R^ is an open convex set in the
PROJECTIVE CONVEXITY
103
affine sense, and the complement has interior points. We have three cases: (a) if ° is contained in two half planes with linearly independent normals. By a suitable choice of coordinates we may assume that xi > 0w and that X2 > 0w in jFf°. Thus ^^ ^ ^ ^ x v ^ %.xx^w ^ ^ ^ J.XX *^*l • ^""'' ^ O,xi/a:o > 0,a:2/xo > 0 or XQ = 0,xiX2 > 0}.
i^i C {(xo : x\ : x^\x^
The line defined by XQ + xi + ^2 = 0 has no point in Ki. If we put this line instead at infinity, then Ki is a bounded, hence compact, convex subset of R^, which means that Ki is a bounded convex subset of R^. (b) KI is a half plane, say KI
= {(xo
: xi
: X2)\XQ
> 0 , X I > 0}.
Taking X2 = 0 as the line at infinity we see that if° consists of two opposite angles, including the interior of the limits at infinity. Hence Ki consists of one of the two open components of the complement of two intersecting lines, together with an interval on each of the lines. (c) KI is bounded by two parallel lines, say KI = {(xo : xi : X2); 0 < Xi
3 and that the theorem has been proved for lower dimensions. Assume at first that d i m L i > 0. Take a point XQ G L I and a plane P{W) C P{V) of codimension one with XQ ^ P ( l ^ ) . The projection K2 C P{W) defined in Lemma 2.5.8 has interior points and is compact and projectively convex. Let L[ = LiH P{W), and let L2 be the projection of L2 in P{W) from XQIt is clear that L[ is a maximal subspace of P{W) contained in the open set P{W) \ K2, and that L2 C K2' By the inductive hypothesis there is a subspace M D L'2 oiP(W) such that M C K2 and d i m X i + d i m M = n2. If N is the subspace of P{V) spanned by XQ and M , then AT D L2? so L2 is maximal in ^2fliV. In KiON the point XQ is maximal since every line in TV passing through XQ contains a point in K2. By the inductive hypothesis it follows that dimZ/2 = dimiV — 1 = d i m M , and since d i m L i = 1 h dimL^, we obtain d i m L i + dimL2 — n — 1. We must also consider the case where d i m i i = 0. Then we have dimL2 > ^ — 2 > 1, for otherwise we could choose a plane of dimension < n — 1 through L i , L2 and an interior point of K2 and conclude by the inductive hypothesis that L2 is not maximal even in that plane. Now we choose XQ G Z/2 instead and form with P ( l ^ ) ^ XQ the projection K i , which is an open set. It is clear that L2 = L2 H P{W) is maximal in P{W) \ Ki. If P ( l ^ ) \ Ki has interior points and dim 1^2 = n — 2, it follows from the inductive assumption that there is a line M C Ki containing the projection L'l of L i . In the twodimensional plane N spanned by XQ and M , we have an interior point Li of A/" fl iiTi, and XQ is a maximal point in N r\K2 since
PROJECTIVE CONVEXITY
105
every line in N through XQ intersects Ki. li N H K2 has an interior point we conclude that N f) Ki contains a line. If AT fi if2 has no interior point, it is a proper subset of a line, and again there is a line contained in the complement NdKi^ so Li was not maximal. It remains to discuss the case where P{W) \ Ki is contained in a subspace H of codimension 1 in P{W) and L'2 = i>2 n P(W) has codimension 2 in P{W). Since L'2 is maximal we know that P{W)\Ki is not equal to ff, so we can choose a line M C P{W) intersecting H only at a point in Ki. Then it is contained in i f i , and we can argue as before to show that Li was not maximal. The theorem just proved is of course also valid with KI replaced by Ki and K2 replaced by K2. Note that it follows from the theorem that if (2.5.2)
n = dimF(y),
n^ = max dimL,
n° = max dimL,
LcKi
LCK°
then every maximal subspace of Ki has dimension n^, and every maximal subspace of if ° has dimension n^. Moreover, (2.5.3)
0 < n° < rij, 2 = 1,2;
ni\n^=n\\n2
— n — 1;
hence n i — n° = n2 — ^2 — z^, where we shall call v the defect If we extend a maximal plane 11^ C K^ to a maximal plane 11^ C Kj, then (2.5.4)
dim(ninn2) = v  l .
(Recall that we have defined the dimension of the empty set to be —1.) In fact, n 4 d i m ( n i n 112) > dim Hi + dim 112 = n i + 77,2 = n — 1 + z/, hence d i m ( n i Pi 112) > z^ — 1. On the other hand, since IIi fl 112 — 0? we have d i m ( n i n 02) + d i m n 2 < d i m n 2  1, for Hi n 112 and II2 are disjoint and contained in 112, which means that d i m ( n i n 112) < i/  1 and proves (2.5.4). All values of the numbers Ui^n^ satisfying (2.5.3) can occur. For let A;,/ be nonnegative integers with A: + / < n — 1, and let k
g(x) = 5 ^ a ; 2 _ 0
k+l+l
^ ^ 2 ^ fc+1
x = (a;o,...,a;„)€R"+\
106
11. CONVEXITY IN A FINITEDIMENSIONAL VECTOR SPACE
The image Kj in P £ of the set where { — lyQ of the set where (1)^(3 < 0. It is clear that
< 0 has as closure the image
nl > k, n 2 > / , n i > A :  h n —fc— / — l = n — 1— /, n2 > n — 1 — k, and by (2.5.3) equality must hold in all these inequalities. We shall prove later that if min(nj,n2) > 0, that is, max(ni,n2) < n — 1, then this is not only an example but equivalent to the general case. However, we shall first discuss the case where ni = n — 1 or n2 = n — 1, which is close to affine convexity and can be studied by a small modification of the proof of Theorem 2.5.7. Assume that n2 = n—1. If we choose the plane at infinity as a subset of K2 in P{V) — P ^ , then K\ is a convex subset of R'^, not equal to R'^. Let k be the maximum number of linearly independent normals of supporting planes. By a suitable affine coordinate change in R'^ we can arrange that Kl C {(a;i,...,Xn);^i > 0, ...,a:ifc > 0}, and K\ is then the intersection of half spaces of the form k
{a;GR^;^a;,Ci e^^''^\x'^D^ip{x)\ < oc,
Va,^.
This implies that e^(^) < C ( l +
\x\)''^J^'^\
where / is the Legendre transform of / . Hence the FourierLaplace form of (^, (2.6.2)'
^ ( 0 
/e^f^'^VW^^,
is defined in {( G C^; /(ImC) < oo}. Since {idldiY{i the FourierLaplace transform of x^D^ip, we have (2.6.4)
sup\ef^'^\^
trans
+ ir]f{d/dO'"^{^'hiv)\
+ ir]Y(p{i + iv) is < oo,
Va,/?.
Here e~^^'^\^ + ^v)^{9/d^)'^(f{^ + irj) should be interpreted as 0 if f{rj) = f oo, and the absolute value is then an upper semicontinuous function in C"^. The convex set M = {rj ^ R'^;/(r/) < oo} may not have interior points, so in general we cannot differentiate with respect to rj. An example is the case / = 0 where Sf is the standard space S, and the FourierLaplace transform is the Fourier transform, defined only in R'^. The aflSne space spanned by M is of the form {a} + MQ, where MQ is a linear subspace of R'^. The fact that / = Hoo outside {a} + MQ means that {x,r]) = {x,a)^ a x e MQ and /(?/) < oo, so f{x) — {x,a) is then constant along MQ by Theorem 2.2.4, that is, it is a function in the dual space IV^/MQ of M Q . Conversely, this implies that / = 4oo outside {a} + MQ. NOW we have the CauchyRiemann equation (2.6.5) _ (t, d/dOip{^ + iv) = W, d/di + id/d7])ip{( + zry) = 0, te MQ, V ^ M\ where Af° is the relative interior of M . This follows at once from the uniform convergence of (2.6.2)' in any compact subset of R ' ^ + i M ^ . We can then differentiate with respect to r/ also in (2.6.4) provided that derivatives are only taken in directions in MQ.
CONVEXITY IN FOURIER ANALYSIS
113
T h e o r e m 2.6.1. Let f be a convex lower semicontinuous function in R'^ which is finite in some open set. Then the FourierLaplace transformation (2.6.2)'is an isomorphism of the space Sf, defined by (2.6.1)', on the space of functions (p in R"^ + zM, M = {rj £ R'*; /(r/) < oo} satisfying the following conditions: (i) {d/d(,)^(p{^ f IT]) is continuous for all a when r] is in the relative interior M° of M. (ii) IfL is any complex line then (fii+irj) is continuous in Lfl (R^ jiM) and analytic in the interior. (iii) The absolute value
is upper semicontinuous
and bounded in C^ if defined as 0 when
f{7]) = fCX).
The inverse is given by (2.6.3)'
ifix) = (27r)^ /e^(^(^ + i77)de is defined and e^*'^^(^^ G S. In view of (iii) we have (2.6.6) e('^^'i)fM\x''D^iPr,{x)\
= e^'Wx"(jD + 277)^e,(^) 2 throughout this chapter.) If X G R'^ \ {0} ^hen the reflection x' of x in the sphere OBR is defined by x' — i? x/rr . It has the same direction as x, and \x\\x' I = i?2. Note that the inversion x \^ x' is the identity on the sphere OBR, and that it is an involution, that is, {x')' = x for every x ^ 0. If x,x' and y,y' are two pairs of corresponding points, then the equation \x\\x'\ = \y\\y'\ shows that the triangles O^x^y and O^y'^x' are similar; we may interchange x and x' in this conclusion. Hence
(3.1.4)
\y\/W\^\x\/\y'\
=
\xy\/\x'y'\,
(3.1.5)
\y\/\x\^\x'\/\y'\
=
\x'y\/\xy'\.
HARMONIC FUNCTIONS
119
If \y\ = R then y' = y and (3.1.4) gives \x  y\/\x' y\ = y/\x\J\x^\ = \x\/R, which means that the sphere is harmonic with respect to x and x'. Now we define the Green's function (cf. (1.5.1)) (3.1.6) GR{x,y) = E{x y)E{{x'  y)\x\/R) = E{x  y)  E{{x y')\y\/R), when x^y E Bn and x ^ y. Here the second equaUty follows from (3.1.5). The first expression shows that GR is a harmonic function of y for fixed X ^ y^ and the second expression that it is a harmonic function of x for fixed y ^ X. In fact, GR{x^y) = GR{y^x). We have (3.1.7)
GR{X, y) = Oii \x\ =Roi
\y\ = R; and GR{X, y) < 0,
since for fixed x with x < R the inequality x — ?/ < \x\\x' — y\/R is satisfied for all 7/ in a ball with OBR on its boundary, by (3.1.4), so it is equal to BR. Uue C^CBR) satisfies (3.1.3), then (3.1.8) u{x) =
GR{x,y)f{y)dy\JBR
dG{x,y)/dnycp{y)
dS{y),
x e BR,
JdBn
where Uy is the exterior unit normal y/R, and dS is the Euclidean surface measure. If n G CI{BR), then (3.1.8) follows from (3.1.2)', for
/
E{{x'y)\x\/R)/lu{y)dy
=Q
because BR 3 y ^^ E{{x' — y)\x\lR) is harmonic. On the other hand, if u vanishes in a neighborhood of the point x, then GR is a harmonic function of 2/ in a neighborhood of the support of u, and
/
{GR{x,y)Au{y)
 {AyGR{x,y))uiy))
dy
JBR
L
dGR{x, y)/dnyu{y)
dS{y)
dBR
by the GaussGreen formula and the fact that GR{X, V) = ^ when y G OBR. By means of a partition of unity we can write t6 as a sum of two functions satisfying one of the preceding conditions, and this proves that (3.1.8) holds iiu E C'^(BR). Conversely, given / G C{BR) it is clear that the first term ui in (3.1.8) is continuous in BR and vanishes on OBR, and it follows from (3.1.2)'' that Aui = f in BR in the sense of distribution theory. (It is not always true that u G C^{BR), but we ignore this point for the moment.
120
III. SUBHARMONIC FUNCTIONS
which is an advantage from using distribution theory.) The second term U2 in (3.1.8) is harmonic in BR since dG{x^y)/dy is a harmonic function of X G BR when y G OBR. We shall prove in Theorem 3.1.5 below that U2 is continuous with boundary values if for every tp G C{dBR), but first we shall calculate dGR{x^y)/dny explicitly. To calculate dGR{x,y)/dy we use the first expression in (3.1.6) which gives CndGR{x,y)/dy
= {y  x)\y  x   "  {y  x')\y  x r ^ W I ^ I ) " " ' 
If we use that \y — x'\/\x — y'\ — \y\/\x\ — i?/x, x' — i?^x/xp and y' = y then this formula can be simplified to CndyGRix, y) = \y x\^{y
 x  {\x\IR)\y

R^xl\x\^))
=
y\yx\^{\\x\^lB?).
Hence we obtain dG{x,y)/dny
= {Rcn)\R'
 \x\^)\y ~ x p " ,
y = R.
We introduce a notation for this Poisson kernel when i? = 1, (3.1.9)
P{x,y)
= cHl\x\')\yx\",
\y\ =
l,\x\ 0, da2 > 0, so da,dai,da2 are multiples of the Dirac measure at y then. Remark. Corollary 3.1.9 means that the set TL^{BR) of positive harmonic functions u in BR with u{0) = 1 is a convex subset of C{BR) with extreme points CnP{/R,y)', these are called minimal positive harmonic functions. By Exercise 3.1.2 HJ^{BR) is compact. Thus Theorem 3.1.8 can be considered as an infinitedimensional analogue of Theorem 2.1.9, with Tij^{BR) forming an infinitedimensional simplex. The Choquet theory referred to after Theorem 2.1.9 allows one to deduce results like Theorem 3.1.8 from a direct proof of results like Corollary 3.1.9. We shall now for harmonic functions in a half space prove analogues of the preceding results on harmonic functions in balls. As a preliminary we shall discuss some further properties of the inversion x H> R^XI\X\^ that will also explain the formula (3.1.6) above for GR. Note that the second term there can be written {\X\IRY~'^E{X' — y) if n > 2, E{x' — y)\(27r)~^ log(x/i?), if n = 2. That it is a harmonic function of x was proved above by appealing to another version of (3.1.6), but it is also a special case of the following theorem. (For the sake of simplicity we take JR = 1 in what follows.) T h e o r e m 3.1.10. Let X C R*^ he an open set, and let X be the reflection {x elU^X {0};x' = x/lx]"^ G X}. Then (3.1.17)
\dx'\ =
\dx\/\x\\
so that X \^ x' is conformal with magnification  x '  /  j ;  . Ifu is a harmonic function in X and u{x) = \x\^''u{x/\x\^),
factor l/\x\'^ — \x'^ —
xeX,
126
III. SUBHARMONIC FUNCTIONS
then u is a harmonic function in X. Proof, Since dXj = dxjl\x\^ — 2xj 'Yj^k dxkl\x\^^ we have rz;pda:;' = rxdx where rx is the reflection in the plane orthogonal to x, hence orthogonal. This proves (3.1.17). Since x and Uj{x) = X j \X\ a r e harmonic functions in R^ \ {0}, it is clear that u is harmonic if ?/ is a firstorder polynomial. A quadratic form q{x) — YTi,k^\ Qjk^j^k is harmonic if and only if Y^l Qjj = 0, and then we have n
q{x) = \x\~''~'^ ^
n
QjkXjXk = Yl Qjk{duj/dxk 
j,k=l
6jk\x\~'^)/{n)
j,k=l n
= 51
Qjkduj/dxk,
j,k=i
so q is harmonic. Thus Au = 0 if li is a harmonic polynomial of second order. If we apply this to the Taylor expansion of ti at x', it follows that Au{x) = 0. The proof is complete. Remark. If n = 2 it is well known that for every conformal map X 3 X H^ (p{x) G X, that is, for every analytic or antianalytic function from X C C to X C C, the composition uocp is harmonic in X if i^ is harmonic in X. The conformality means that V ' is proportional to an orthogonal matrix, that is, dicpi = ±d2(p2,
^2^1 =
Tdi(p2,
hence Acpi = ±8182^2 T ^2^1 2 every conformal map (^ is a product of inversions and orthogonal transformations (see Berger [1, p. 223]), so it follows from Theorem 3.1.10 that X 3 X ^^ \ip^\~^~{uoip) is harmonic in X if tt is harmonic in X. Exercise 3.1.5. Prove with the notation in Theorem 3.1.10 that {Au){x) = x2^(A^)(x/xn, first when u E C^(X), then when u G V'{X). If X is the half space {x G R'^'.Xn >  } , then X = {x e IC';2xn > xp} is the ball with radius 1 and center at ( 0 , . . . , 0,1). Thus Theorem 3.1.10 gives a linear onetoone correspondence between harmonic functions in the
HARMONIC FUNCTIONS
127
half space and in the ball which can be used to carry our results from the ball to the half space. One should just keep in mind that the boundary point 0 of the ball X corresponds to a boundary point at infinity in the half space X , and this point can carry a positive measure in the analogue of (3.1.15). We have u{x) —> c as x —> 0 if and only if \x\'^~'^u{x) ^ c as a; —> oo in X . Since \x\'^~'^u{x) = u{x/\x\'^), re G X , we see that if u e C\X) then 9^(x^2^(j:))  0(x2«l), a < 1. To relax these rather unnatural conditions we shall repeat some of the arguments for the ball in the case of the half space instead of using the inversion to just carry the results over. We define the Greenes function for the half space H = {x E R"^; Xn > 0} by
(3.i.6y
_
GH{x,y) = E{xy)E{xy'')
= E{xy)E{x^y),
x.yeH,
x^y,
where y* = (?/i,..., ^/ni? —Vn) is the refiection of y in the boundary plane dH. It is clear that Gnix^y) = Gniv^x) < 0 is harmonic in x (in y) for fixed y (fixed x), x ^ y, and that G{x,y) = 0 if x E dH or y E dH. We define the Poisson kernel when x E H and y G dH by (3.1.9)' Pnioc^y) = dGH{x,y)/dyn = 2dE{x  y)/dyn = 2xn\x  yl'^'/cn. (We shall often identify dH with R ^  ^ ) Then the analogue of (3.1.8)' is that (3.1.8)" uix)=
[ GHix,y)Au{y)dy+
[
PH{x,y)u{y) dS{y),
X e H,
JdH
JH
provided that u G C'^{H) and that u{y) and (1 \yn)du{y)/dy are bounded in H. The integral over H is defined as the limit of the integral over {y G JH"; \y\ < ^} as ^ ^ oo. The only difference in the proof is that we have to apply the GaussGreen formula to
/
{GH{x,y)Au{y) 
{AyGH{x,y))u{y)dy
integrated over {y G H; \y\ < g} and then let ^ ^ oo. We must show that the integral
/
{GH{x,y)du{y)/dny

dGH{x,y)/dnyu{y))dS{y)
taken over the spherical part of the boundary converges to 0. The area is CnQ^'^/'^ and dGnix^ y)/dy = 0{\y\'~'^) by the mean value theorem, so the
128
III. SUBHARMONIC FUNCTIONS
second part of the integral is 0(1/g). Since Gni^^y) — 2xn{xn  2/n)^ — y\~^ jCn + 0(1^1"^) the same is true of the first part, which gives (3.1.8)''. In particular, when u — \^e obtain (3.1.10)'
/
PH{x,y)dS{y)^l,
x e H.
We can now prove: T h e o r e m 3.1.5'. Let u be a bounded continuous function in H which is harmonic in H, and let ^{y') = u{y',0), y' G R'^"^. Then we have (3.1.1iy
u{x)=
I PH{x,y')^{y')dy', JdH
x e H.
Conversely, if cp is a given continuous function in R^~^ such that the integral J \ip{y')\{l H \y'\)~'^ dy' is finite, then (3.1.11)' defines a harmonic function in H which is continuous in H, u{y'^0) = ^{y')Proof. Since u is harmonic we know that u G C^{H)^ and ii \u\ < M in H then yn\du{y)/dy\ < 2nM in i?, by (3.1.14)' applied to S i 9 a; H^ M =b u{y + ynx). Hence we may apply (3.1.8)" to u{x',Xn + 6:) if e > 0, which gives u{x',Xn^e)=
/
PH{x,y')u{y',e)dy'.
JdH
In view of (3.1.10)' and the boundedness of u we obtain (3.1.11)' when e —» 0. The proof of the strong converse is essentially a repetition of the corresponding part of the proof of Theorem 3.1.5, so it is left for the reader. Next we extend the RieszHerglotz theorem. Note that x i^ Pnix^y) for y G dH is clearly a minimal positive harmonic function for the boundary point y, for it vanishes at all finite boundary points ^ y and \x\'^~'^PH{x,y) ^ 0 as a; ^ oo. For the point at infinity we have the minimal positive harmonic function which vanishes at every finite point of the boundary. It is therefore clear that Theorem 3.1.8 must take the following form: T h e o r e m 3.1.8'. Let u be a harmonic function which is nonnegative in H. Then there exists a unique positive measure da on dH = IV^~^ and a constant a > 0 such that (3.1.15)'
u{x) = axn+
/ PH{x,y')d(j{y'), JdH
x e H.
HARMONIC FUNCTIONS
129
We have / ( I I \y'\)~"^ da{y') < oc, da is the weak Umit of the measure u{x\ Xn)dx' as Xn ^ 0, and a is determined by the behavior ofu at infinity in the sense that u{tx)/t ^ axn in L\^^{H) as t ^ f oo. Moreover, (3.1.18)
/ ( I + \x'\^)^u{x\xn)
dx'
< ( 1  f Xn) / (1 + k T ) ~ ^ da{x')
and for every ^p G C{W^)
/
+ \CnaXn,
Xn > 0,
with (p{x') = 0 ( ( 1 +  x ' p )  t ) we have
ip{x') da{x') = lim /
ip{x')u{x'^Xn)dx'.
Proof. As already indicated we could obtain Theorem 3.1.8' from Theorem 3.1.8 by means of an inversion, but we shall give another proof to avoid some computations. First we assume that u is continuous in i J , and we write down (3.1.11) for the ball {x G R"^; xp < 2Rxn} with radius R. This gives for every x ^ H when R is large enough u{x) = {2xn  \x\yR)c'
J \ x  y\My)
dS{y)
with the integral taken over the boundary sphere. When R ^ oo we obtain by just keeping the integral where yn = \y^\'^/{R + ^/R^~^W\^) ^^^ v' i^ in a compact set u{x) > /
PH{x,y)u{y)dS{y)
= v{x),
x e H.
JdH
Thus u{y)/{l + 1^1)"^ is integrable over dH^ so it follows from Theorem 3.1.5' that t; is a continuous function in H with boundary values equal to those of It, and that v is harmonic in H. Hence u — v is a nonnegative harmonic function which is continuous in H and has boundary values 0. The inversion argument shows that all such functions are multiples of the minimal positive harmonic function x H> x^ corresponding to the point at oo. Hence we obtain u{x) — v{x) = aXn for some a > 0, which proves (3.1.15)' when u is continuous in H. To extend (3.1.15)' to general positive harmonic functions in H we apply the result already obtained to u{x',Xn + s) which gives u{x)=
/ JdH
P{x\xne,y')u{y\e)dy'\a^{xn~e),
Xn>e.
130
III. SUBHARMONIC FUNCTIONS
For 0 < £ < 1 it follows that 0 < a^ < C, / ( I + Wiyuiy'.e) dy' < C. We can therefore choose Sj ^ 0 such that u{y'^ej)dy' converges weakly to a limit d(j{y'), and we have / ( I + \y'\)~'^dG{y') < oo. When 0 < e < 1 and \x'\ < i? > 1 , we have P{x', Xn  £, y') u{y\ e) dy' < C'{xn  e), / \y'\>2R for (1 + \y'\)l\x'  2/' < 3 when \x'\ < R< Poisson integral v{x)^
f
2/', so it follows that for the
P{x,y')da[y')
JdH
we have v{x) < u{x) and u{x) — v{x) < Cx^ in H. Hence u{x) — v{x) is a minimal positive harmonic function for the point at infinity, so u{x) = v{x) + aXn: which proves (3.1.15)'. Now we obtain u{x'^Xn)dx' —> da{x') as Xn ^ 0, for ii cp e (7o(R'^) then / u{x'^Xn)^{x')dx'
= axn I (p{x')dx' \ I da{y')
/
P{y\xn')X')ip{x')dx'
where the inner integral converges uniformly to 0}, and Re/(2;) > 0 there, then
/(.) = i(a. + 6) + i / ; j (1^ + j l p ) Mi). where dcr > 0, a > 0, & G R, and / da{i)j{l
+ i^) < CXD.
E x e r c i s e 3.1.7. Show that if ?x is a positive harmonic function in if, then either t'^~^u{tx) ^ oo as t ^ oo, uniformly on every compact subset of i7, or else a = 0 and ^Qjjda{y') < oo in the representation (3.1.15)' of 16, and then we have t'^~^u{tx) ^ PH{X^O) J^jjda uniformly on every compact subset of H. We shall now discuss the extension of Theorem 3.1.8' to functions which are not semibounded. To obtain representations of the form (3.1.15)' it is clear that one must impose restrictions both as x^i —> 0 and as x ^ cx), where we have to rule out functions like nx^ — I^P which is bounded above when Xn is bounded but grows too fast with Xn (We shift to upper instead of lower bounds now to conform with the analogous discussion in Section 3.3.)
132
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3 . 1 . 1 1 . Let u be a harmonic function in H such that u~^ = max('u, 0)
(3.1.20)
lim
/ u'^ix', Xn){l + \x'\y
(3.1.21)
lim
/
with
dx' < oo,
u^{x)dxlBJ' + 1 < 0 0 .
Then there is a unique measure da with / ( I j 2/')~'^ \da{y')\ < oo on dH = R^~^ and a constant a such that (3.1.15)' is vaUd. da is the weak limit of the measure u(x',Xn)dx' as Xn —> 0, and a is characterized by the behavior of u at infinity in the sense that u{tx)/t ^ axn in L\Q^{H) as t ~> +00. Moreover, (3.1.180
j{l^\x'\^)'^\u{x',xn)\dx' 0,
and for every ip G C ( R ^  ^ ) with ip{x') = 0 ( ( 1 + l^r'p)?) we have
/
ip{x')da{x')
— lim Xn^^
/
J
ip{x')u{x',Xn)dx'.
Proof. So far we have only used the Green's function and the Poisson kernel for the ball and for the half plane. However, for the half ball S ^ = H n BR we can also easily define a Green's function by G\{x,y)
= GR{X,y)GR{X,y*)
= GR{X,y)GR{x%y),
x,y e B^,x^
y,
where * denotes reflection in dH just as in (3.1.6)'. In fact, it is clear that G%{x^y) = 0 if X or y is in d B ^ , and G\{x,y) — E{x — y) is harmonic in X and in y. The Poisson kernel Pt{x,y)
= dG+{x,y)/dny,
x E B+, y € dB+
is positive, for G^{x,y) < 0 in S j x B j by the maximum principle since G^{x,y) —>  o o as a; ^ y, and Green's formula gives as before if v is harmonic near BR (3.1.11)"
v{x)=
f JdBt
P^{x,yHy)dS{y).
HARMONIC FUNCTIONS
133
If 7/ G B^ then the harmonic function G\{x^y) — Gnix^y) of x E B^ is > 0 when x G dB^^ so it follows that G j > GH in 5 ^ x 5 J , which proves that (3.1.22)
PR^X.V)
< PH{x,y),
iix e B^ and y G
dB^ndH.
When X G J5j and y G 5 5 J \ 9iJ we have by the mean value theorem (3.1.23)
P^{x, y) = R'^iP{x/R,
= R{1  \x\yR')c'{\x
y/R)  P[x^lR,
 y\ \x
yTn
y/R))
< ^nXn{R 
\x\)^c\
By (3.1.20) we can take a sequence £j —> 0 such that the integral Ju'^{x',ej){l + Ix'l)"^ dx' has a finite limit as j —> oo and the sequence u'^{x'^ej)dx' converges weakly to a measure dv > 0, necessarily with / ( I h \x'\ydiy{x') < oo. If we apply (3.1.11)" to u{x',Xn + Sj) and let j ^ oo after integrating with respect to i? for ^ < i? < 2^, it follows that for large g gu{x) < f
dR [
JQ
P^{x,y)u^{y)dS{y)^Q
JHndBR
[
PH{x,y')dv{y').
JdH
Here we have used that u'^ G Ll^^{H). Hence we obtain using (3.1.23) (3.1.24)
u{x) o. Choosing T'(0) = —aid as we may, we conclude that (3.1.35) ^ ( ^ ) ( 0 ; e , e , y , y )  6au''\0;e,Y,Y) \Ga\'\0]Y,Y)
> 0,
Y G R^.
The lefthand side is a harmonic quadratic form, for u is harmonic, and since it takes its minimum at 0 it must vanish identically. Returning to (3.1.34) we can choose T so that T{0)Y = aY\eZ for any Z with (Z,'u'(0)) = 0, and since there is equality in (3.1.34) when ^ = 0, the coefficient of £ must vanish, that is, 47i'"(0; e, Z, Y)  8au''{0] Z,Y) = 0
HARMONIC FUNCTIONS
139
for all Y and Z with {Z,u'{0)) = 0, which implies (3.1.32). Hence the harmonic quadratic form u"'{0] e, F, Y) — 2au'\0] Y, Y) must be a constant times {Y,u'{0))'^, and the harmonicity implies that the constant is 0. This proves (3.1.29) with j = 2. The remaining condition in (3.1.34) is only that 2u"{0; Z,Z)>0 when (Z, u'(0)) = 0, which follows from the fact that u"{0; •, •) is positive semidefinite in the tangent plane of the level surface of u at 0. With T chosen so that T'{0) = —aid we know now that (3.1.31) must vanish of third order, hence of fourth order because of the positivity, and we can repeat a similar argument. However, it is now time to specify the inductive statement which will prove (3.1.29). We shall prove for z/ = 1,2,... that (3.1.29) is valid for 1 < j < z/ + 1 and that for 0 <j 1. Then we can choose the tangent vector field T so that for a given symmetric u\l linear form Z in R'^ with values in the orthogonal plane of u'{0) in R"^ (3.1.30)' T(0) = e,
T'(0) =  a i d ,
T^^\0) = 0, K
j < u,
T(^+I)(0)
= Z.
In fact, the j t h differential of (T(x),'a'(x)) at 0 in the direction Y is
(3.1.37)
Yl (^)^^"^^\0',T^'~'\0;Y^'),Y').
If T has the Taylor expansion required in (3.1.30)' and j < ^' I 1, this reduces to ^(^^^)(0;e,y^)ia'a(^)(0;y^) = 0 by the inductive hypothesis. Thus R{x) = {T{x),u'{x)) = 0{\x\'''^'^), and replacing T{x) by T(x)  R{x)u'{x)/\u'{x)\'^ we obtain all the conditions (3.1.30)'.
140
III. SUBHARMONIC FUNCTIONS
When T satisfies (3.1.30)' and j < 2i/ + 1 the j t h differential of u''{x;T{x),T{x)) at 0 in the direction Y is 31
(3.1.38)
^^^•+2)(0;e,e,y^) + 2 ^
('•^\(^+2)(Q.g^yy0(^
+ i(i  l)^«)(0;T'(0;y),r'(0;y),y^') + 2 J^
^^
^7x(^+')(0; T'(0; Y), T ( ^  ^  ^ ) ( 0 ; y^"^"^), F^)
if j < 2i/ + l, for terms where both "factors" T are differentiated more than once must vanish by (3.1.30)' since one dijfferentiation is of order < u. If j = 2i^ + 2 there is an additional term
Using (3.1.30)' we simplify (3.1.38) for j < 2z/ + 1 to u (^•+2)(0; e, e, Y^)  2aju^^+^\0; e, Y^) J^'i
+2 E 2= 0
.,
7T^^:Tn^^'^'^(0;e,TO)(0;y^^),yO+i(il)aV^)(0;y^^^ ^^
*^**'
Since j — u 1. Thus the inductive hypothesis remains valid with u replaced by z/ + 1, which completes the proof. 3.2. Basic facts on subharmonic functions. Using harmonic functions instead of linear functions we can now copy Definition 1.1.1 for functions of several variables: Definition 3.2.1. A function u defined in an open subset X of R"^ with values in [—cx), oo) is called subharmonic if (a) u is upper semicontinuous; (b) for every compact subset K oi X and every continuous function h on K which is harmonic in the interior of if, the inequality u < h is valid in K if it holds in dK. A function u is called superharmonic
if —u is subharmonic.
Remark. It may seem inconsistent that we require upper semicontinuity here while we imposed lower semicontinuity on the convex functions in Chapter II. However, we only consider subharmonic functions in open sets, and a convex function in an open set is continuous. Since upper semicontinuity means that a strict upper bound valid at one point is also valid in a neighborhood, it is clear that (a) is the natural condition to go with (b). The function u = —oo is subharmonic according to Definition 1.3.2. This is sometimes convenient but some authors exclude this function in the definition, which is convenient at other occasions.
142
III. SUBHARMONIC FUNCTIONS
By Theorem 3.1.6 every harmonic function is of course subharmonic and superharmonic. The most important example of a subharmonic function in R'^ which is not harmonic is a fundamental solution x — t > Ey{x) = E{x — y)^ where E is defined as in Theorem 3.1.2, equal to —oo at 0. It is clear that Ey is then continuous with values in [00,00). If y ^ K then it is also obvious by Theorem 3.1.6 that condition (b) in Definition 3.2.1 is fulfilled, and ii y e K then Ey < h in some open neighborhood V of y^ and since diK \ V) C (dK) U (dV) we have Ey < h in d{K \ V), hence inK\V and in K. In the same way we see that a finite sum a; i> ^ ajE{x — yj) is subharmonic if a^ > 0. In particular, if / is an analytic function ^ 0 in an open connected set X C C, then log \f{z)\ is subharmonic, for if we write f{z) = g{z) H i {z — Zj) with g analytic and ^ 0 in K, then log ^ is harmonic near K and Yli log \^ ~ ^j\ is subharmonic. This example is the reason for the importance of subharmonic functions in analytic function theory. The case n = 2 often diff'ers from the case where n > 2 because the fundamental solution is not negative in the whole space when n = 2, but since many of the most important applications occur when n = 2 we cannot ignore these special features by assuming n > 3. Exercise 3.2.1. Show that if X^and X are open sets in R'^ and O is an orthogonal transformation, X = OX, then X 3 x — t > u{Ox) is subharmonic in X if lA is subharmonic in X . Also prove that X 3 x t^ \x\'^~'^u(x/\x\'^) is subharmonic in X if iz is subharmonic in X and X, X are related as in Theorem 3.1.10. Show that if X, X C C = R^ and cp is a. complex analytic bijection X —> X , then u o (p is subharmonic in X ii u is subharmonic in X. T h e o r e m 3.2.2. If u is subharmonic in X and 0 < c E R, then cu is subharmonic in X. If ui,... ,Uy are subharmonic in X , then u — m a x ( t i i , . . . ,Uj^) is also subharmonic in X. If Ui^, t E / , is a family of subharmonic functions in X and u{x) = sup^^j Ui^{x) is upper semicontinuous with values in [—00,00), then u is subharmonic in X. If ui,U2, • • is a decreasing sequence of subharmonic functions in X , then u = lim_^_,oo '^j is also subharmonic in X. Proof. The first three statements are obvious from the definition. To prove the last one we take a function h which is continuous in a compact set jFf C X and harmonic in the interior of K, such that h > u on dK. Let e > 0. For every 0:0 G dK we have Uj{xo) < h{xo)\e for some j , and since Uj — h is upper semicontinuous there is a neighborhood V of XQ such that Uj{x) < h{x) 46:,
iix eV
nK.
Here we may replace Uj by Uk for any k > j . By the BorelLebesgue lemma we can cover dK with a finite number of such neighborhoods V,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
143
hence Uk(x) < h{x) 4 £ when x G dK^ if k is large enough, and it follows that u < Uk < h\ e m K. Any decreasing limit of upper semicontinuous functions is upper semicontinuous, which completes the proof. Definition 3.2.1 is often useful as it stands, but it does not indicate for example that the sum of subharmonic functions is subharmonic. We shall therefore give other equivalent properties, analogous to those in Exercise 1.1.12 or Theorem 3.1.12. T h e o r e m 3.2.3. Let u be an upper semicontinuous function in an open set X C R"^ with values in [oo, oo). Then each of the following conditions is necessary and sufficient for u to be subharmonic in X: (i) Condition (b) in Definition 3.2,1 is fulGUed when K is a closed ball
ex. (ii) If Xs = {x^X\y£Xif\y {x,r)=
— x\< 6], ^ > 0, then the mean value /
(3.2.1)
M^{x,r) = Mix,r)= f u{x + ry) du{y)/cn, x e Xr, is an increasing function ofrE [0,6] ifx G X^. (iii) For every positive measure dfi in the interval [0, ^], ^ > 0, we have (3.2.2) u{x)
/ duj{y)d^{r) < I I u{x J\y\=i Jre[oM J\y\='^ Jre[o,d]
•^ry)du{y)d^{r),
ifx e Xs. (iv) For every 6 > 0 and every x E X^ there is a positive measure supported by [0,6] but not by {0} such that (3.2.2) is valid. Note that the integrals are well defined since u is semicontinuous. Proof. It is obvious that subharmonicity implies (i) and that (iii) implies (iv). From (ii) it follows that u{x) < M{x,r) ii r < 6 and x G Xs, hence (ii) implies (iii). To prove that (i) implies (ii) let x £ Xs and set K = {y] \x — y\ < 6}. If (^ is a continuous function on the unit sphere such that u{x + 6y) < (p{y) when \y\ = 1, then condition (i) states that u < h in K if h is the solution of the Dirichlet problem given in K by h{x \z)=
P{z/6, y)cp{y) du{y),
\z\ < 6.
I f O < r < 5 t h e n b y (3.1.12) M{x,r)
< / J\z\ = l
h{xhrz)duj{z)/cn
= h(x) = / J\y\ = l
(p{y) duj{y)/cn.
144
III. SUBHARMONIC FUNCTIONS
Since u is upper semicontinuous the infimum of the righthand side over all continuous majorants cp is equal to M{x^6)^ which proves the monotonicity stated in (ii). It remains to prove that (iv) implies that u is subharmonic. We can essentially repeat the argument in the proof of Theorem 3.1.12. Let K be a compact subset of X and let /i be a continuous function on K with h> u on dK, such that h is harmonic in the interior of K. If the supremum V oi V — u — h in K is positive, then it is finite and is attained in a compact subset F of the interior of K, because v is upper semicontinuous. Let XQ E F have minimal distance 2^ > 0 to dK. On every sphere with radius r G (0, 28) and center at x there is some point y at distance 2^ — r to dK, and since v is upper semicontinuous we know that u < V^ in a neighborhood of y. Hence /
v{x + ry) duj{y) < c^V — Cnv{x),
0 < r < 2^.
^12/1 = 1
If we integrate with respect to a measure d\i with support in [0, S\ with the properties in condition (iv), we get a contradiction with the hypothesis that (3.2.2) holds, for the inequality (3.2.2) must also hold with u replaced \yY V — u — h since h is harmonic. This contradicts the assumption that F > 0 and proves that u the lefthand side by (3.2.2), and the upper limit is < the lefthand side since u is upper semicontinuous. E x e r c i s e 3.2.2. Prove that iiu is upper semicontinuous in X then u is subharmonic if and only if lim^_^o(^n(^5^) — u{x))/r'^ > 0 for every x G X with u{x) > —oo. (Hint: Prove first that u{x) + £a:p is subharmonic.) Prove that for every subharmonic function u in X and x E X lim /
\u{x + y) — u{x)\ dy/r'^ = 0, lim /
if u{x) > —oo,
\u{x 4 y)\ dy/r"^ = CXD, if u{x) = —oo.
(Hint: Examine J\y\^^ \u{x iy) — c\ dy/r"^ when c > u{x).)
BASIC FACTS ON SUBHARMONIC FUNCTIONS
Corollary 3.2.4. u is both subharmonic and superharmonic ifu is harmonic. Corollary 3.2.5. If ui,... subharmonic.
,Uk are subharmonic
145
if and only
then ui \  •  \ Uk is
Corollary 3.2.6. Ifu is a function defined in an open set X such that every point in X has a neighborhood in which the restriction of u is subharmonic, then u is subharmonic in X. Thus subharmonicity is a local property. Corollary 3.2.7. Let Y C X be open sets, and assume that u is a function in X which is harmonic in Y and equal to —oo in X\Y. Then u is subharmonic if and only if u is upper semicontinuous, that is, u{y) ^ —oo ifY3y^xeX\Y. Proof. Note that (3.2.2) is trivial ii x E X \Y and follows for small 6 from the mean value property of harmonic functions ii x EY. Corollary 3.2.8. Ifu is subharmonic in an open connected set X and not = —oo, then u E Ll^^{X), so u(x) > —oo almost everywhere. Proof. li X £ Xs^ defined as in Theorem 3.2.3, and u{x) > —oo, then u is integrable in {y; \x — y\ < r } if r < (5, for u is bounded above and the integral is > u{x)r'^Cn/n. The subset Y oi X consisting of points such that u is integrable in some neighborhood is open by its definition, and we claim that it is closed in X . In fact, iiy is in the closure and y E Xs^ then we can choose X E X with a; — ?/ < 6/2 so that u{x) > —oo, and u is integrable in the ball with radius S/2 and center at x, which has y as interior point. Thus Y is open and closed in X , and since X is connected it follows that Y = X or Y = ib, so u E i^ioc(^) oi"^ =  o o . Corollary 3.2.7 shows again that positive linear combinations of fundamental solutions E{ — y) are subharmonic, and that log  /  is subharmonic if / is an analytic function. We can prove this more generally: T h e o r e m 3.2.9. Let du be a positive measure in R'^ with support, and set (3.2.3)
u{x) = j / u{x)
compact
E{xy)dv{y).
Then u is subharmonic; it is called the potential of the measure dv. The convolution E ^ dv in the sense of distribution theory is defined by u. Proof. li t is a constant, then Et = max(£^, ^) is subharmonic and continuous, by Theorem 3.2.2, and Et I E diS t I —oo. Hence the continuous function Ut{x) = {Et * dv){x) = JEt{x — y)dv{y) decreases to u{x) as
146
III. SUBHARMONIC FUNCTIONS
t [ —oo, which proves that u is upper semicontinuous. To prove that u is subharmonic it suffices by Theorem 3.2.2 to show that Et * dv is subharmonic, which follows from Theorem 3.2.3: / ut{x + ry) du;{y) J\y\=i = / diy{z) / J
Et{x'\ryz)duj{y)
J\y\ = l
> Cn
Et{xz)dv{z)
= CnUt{x).
J
(This is really a continuous version of Corollary 3.2.5, for the convolution is a superposition of translates.) Since Et  E ^ 0 in L^ we have Ut — u ^ 0 in L^, as t —> — oo, and since convolution is continuous in the distribution topology it follows that u = E ^ dv. The proof is complete. Remark. When n > 2 the fundamental solution is negative everywhere so (3.2.3) is well defined with values in [—00,00) even if dv does not have compact support. Since u is the decreasing limit of the same integral taken only for I2/I < jR when i? ^ 00, it follows that u is subharmonic also in this more general case. However, we may have u = —00 then. Theorem 3.2.9 allows us to give a simple example of a subharmonic function which is not continuous with values in [—00,00). We just take sequences xjt ^ 0 in R"^ \ {0} and a^ > 0 with YlT ^kE{xk) = —1. The sum u{x) = YlT ^kE{x — Xk) is subharmonic, u{xk) = —00 for every fc, but u{0) = —1. lit < 1 then Ut{x) = msix{t,u) is subharmonic and takes values in [t,oo), ut{0) =  1 but lim^_^Qn^(x) = t < —1 = ut{0). To establish a connection between subharmonicity and Poisson's equation we begin with an elementary result similar to Corollary 1.1.10: P r o p o s i t i o n 3.2.10. Ifu G (7^(X) where X is an open set in R^, and M is defined by (3.2.1), then (3.2.4)
lim(M(x, r)  u{x))/r'^ = Au{x)/2n,
x e X,
r—vO
and u is subharmonic
in X if and only if Au > 0.
Proof. It suffices to prove (3.2.4) when rr = 0. By Taylor's formula n
u{x) =u{0)^Y^Xjdju{0)i^ 1
n
^
XjXkdjdku{0) + R{x),
R{x) = o{\x\'^).
j,k=i
Hence /. ,^R{ry)duj{y)/r'^ —> 0 as r ^ 0. The integrals over dBr of the terms in the sums are all zero apart from
BASIC FACTS ON SUBHARMONIC FUNCTIONS
147
which proves (3.2.4). Here we have used that /. , ^ 7/? da;(y) = Cn/n since the integral is independent of j . Prom (3.2.4) and Theorem 3.2.3 it follows at once that A?x > 0 if u is subharmonic, and that u is subharmonic if An > 0. If we just assume that Aii > 0 we may conclude that u{x) + e\x\'^ is subharmonic for every 6 > 0, and when ^  0 it follows that u is subharmonic. We could also prove the second part of Proposition 3.2.10 using an analogue of Corollary 1.1.16: P r o p o s i t i o n 3.2.10'. Ifu is an upper semicontinuous function in the open set X C R'^ which is not subharmonic, then one can find XQ ^ X and a quadratic polynomial q with Aq < 0 such that q{xQ) = u{xo) and u < q in a neighborhood of XQ. Conversely, u is not subharmonic if there is such a function q. Proof. The last statement is obvious, for x i> u{x) — q{x) — £\x — rrop would be subharmonic for small ^ > 0, equal to 0 at XQ but < 0 in a punctured neighborhood of XQ. Now assume that u is not subharmonic. We can then choose a closed ball B C X and a function h which is continuous in B and harmonic in the interior of B such that u — h0. Set for 5 > 0 Ve{x) = h{x) — £a:p. Then u < v^ on dB if e is sufficiently small, but sup^(?z — Ve) > 0. The upper semicontinuous function u — Ve takes its maximum in B at an interior point rro, so u(x)
< Vs(x)
h u(Xo)
— Ve{xo),
X £
B.
Set q{x) = u{xo)  Ve(xo) + ^
d'^h{xo){x  Xo)'^/a\  £:x^ + ^€\x  Xo^
a 0. It is easy to extend Proposition 3.2.10 to distributions:
148
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3.2.11. Ifu is a subharmonic function in an open set X and is not = —oo in any component, thus u G L\Q^{X), then Atfc > 0 in the sense of distribution theory. Conversely, ifU G V'{X) and AC/ > 0, then U is defined by a unique subharmonic function u in X. Proof. That Ait > 0 in the sense of distribution theory means by definition that fuAvdx
> 0,
live
C^{X),
v>Q.
If we express Av using (3.2.4) we obtain since (3.2.4) is uniform in x — / uAvdx 2n J
— hm / u{x){My{x,r) r^o J
— v{x))/r^
dx
= hm / v{x){Mu{x,r)
— u{x))/r^
dx.
The righthand side is nonnegative by Theorem 3.2.3 iiu is subharmonic, which proves the first statement. To prove the second statement we choose (f G CQ°{BI) such that cp > 0 and (p{x) only depends on \x\, J (p{x) dx = 1. Set ipeix) =:= e''(p{x/s). Then Ue = U ^ ips e C ^ ( X ^ ) , e > 0, where Xs is defined as in Theorem 3.2.3, and AUe = (AC/) * (^^ > 0, so C/^ is subharmonic by Proposition 3.2.10. Hence it follows from Theorem 3.2.3 that U ^i^e^ips, which is defined in X^^s, is a decreasing function of 8 as 5 I 0. Letting e ^^ 0 we conclude that C/ * (^^ is a decreasing function of ^ as 5 I 0, and then it follows from Theorem 3.2.2 that U ^ (ps I u, where u is subharmonic and not identically — oo. Since U ^ (ps ^ U in V^ and U ^ (fs —^ u in LJQ^, it follows that the distribution U is defined by the function u. For any subharmonic function u we have t6 * 0, so the subharmonicity of u is also a consequence of Theorem 3.2.11. If / is an analytic function ^ 0 of a complex variable z, then A log 1/(2:)! is the measure
2TE
rrijdz
where Zj are the zeros and ruj their multiplicities. In fact, in a neighborhood of Zj we have f{z) = [z  Zj)'^^g{z) where g is analytic and ^ 0, so log 1/(2^)1 = 27rmjE{z  Zj) ^ log \g{z)\ where the last term is harmonic. Extending the result on harmonic functions in Exercise 3.1.2 we shall now prove:
BASIC FACTS ON SUBHARMONIC FUNCTIONS
149
T h e o r e m 3.2.12. Let Uj be a sequence of subharmonic functions in an open connected set X C R^, which have a uniform upper bound on every compact subset of X. Then either Uj ^ —oo uniformly on every compact subset of X, or else there is a subsequence Uj^ which converges in Ll^^{X). Ifuj ^ —oo for every j and Uj ^ U in V, then U is deRned by a subharmonic function u and Uj ^ u in Ll^^{X). Proof. It suffices to prove this with X replaced by a relatively compact subset, so subtracting a constant we may assume that Uj < 0 for every j . If Uj does not converge to — oo uniformly on every compact set, then we can find jk and Xk such that all Xk belong to a compact subset K oi X and Ujj^{xk) is bounded. We may assume that Xk ^ XQ E X, and to simplify notation we assume that jk = k. By Corollary 3.2.8 we have Uj E L\^^{X) for every j . If B C X is a closed ball with center at a;o, then the sequence Jg Uj is bounded from below. In fact, for large j there is a ball Bj with center at Xj such that B C Bj C X^ and then we have / Uj dx > JB
Uj dx >
m{Bj)uj{xj).
JBi
We can now show as in the proof of Corollary 3.2.8 that if Y is the set of points X e X having a neighborhood N such that the sequence J^ Uj is bounded from below, then Y is both open and closed, hence equal to X. This proves that the sequence Uj is bounded in L J Q ^ ( X ) . We can therefore find a subsequence Ujj^ which converges in the weak topology of measures, hence as a distribution, and the limit is defined by a subharmonic function in view of Theorem 3.2.11. It remains to prove the last statement, so assume now that Uj —» U in D'. Then AU = lim^_,oo AUj > 0, so [/ is defined by a subharmonic function u^ by Theorem 3.2.11. With the notation in the proof of Theorem 3.2.11 we have (3.2.5)
% ( ^ ) ^ %• * y^si^) ^ u^ ^six),
X G Xs,
and the convergence here is uniform on compact sets in Xs since the convolutions Uj * ips{x) are equicontinuous there. Choose x ^ 0 in CQ^{XS) and e > 0. Then / {u^^s{x)+eUj{x))x{x)
dx ^ / {u^(ps{x)\eu{x))x{x)
dx,
and the integrand on the left is positive for large j . Hence lim / \u — Uj\xdx 3^ooJ
u in L^^^{X) for p G [ l , n / ( n — 2)), while Uj —> u' in L\^^ for p G [1, n/{n  1)). For every x e X we have (3.2.6) More generally^ ifK (3.2.7)
iim Uj{x) < u{x),
x e X.
is a compact subset of X and f G C{K),
then
Iim sup{uj  f) < s\ip{u  / ) .
If da is a positive measure with compact support in X such that the potential E ^da in Theorem 3.2.9 is continuous, then there is equality in (3.2.6) and u(x) > —oo for almost every x ^ X with respect to da. Moreover, Ujda —^ uda in the weak topology of measures. Proof. We keep the notation in the proofs of Theorems 3.2.11 and 3.2.12. From (3.2.5) it follows at once that we have a uniform upper bound for Uj on any compact subset of X , and that Iim Uj{x) < u^ ^si^)^
X ^ Xs
The righthand side converges to u{x) as (5 ^ 0, which proves (3.2.6). Let M — snpj^{uf). UK C Xs a n d x G K, thenmax{M,u*(ps{x)f{x))  M when ^ I 0. It follows from Dini's theorem that the convergence is uniform on K, which proves (3.2.7). If Et = max(J5, t) for some large negative t, as in the proof of Theorem 3.2.9, then Et is continuous with finite values and E — Et —^ 0 in L^ SiS t ^  o o for every p G [ l , n / ( n  2)). Let F ^ X and let 0 < x ^ C^iX) be equal to 1 in Y. The positive measures dfij — tlUj converge to d[i = Au in V^, hence in the weak topology of measures. If we set duj = X^/^j ^^^ dp = xcf/x, then Uj = E ^ duj \ Vj,
u = E ^ du hv,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
151
where A'^j — (1 — x)dlJ^j = 0 in y and Vj —> y in V\X) as j —> oo, hence Vj ^ ' i; in C^{Y) by Corollary 3.1.4. Hence it suffices to examine the convergence of B * duj to E ^ du. We have \\{E  Ft) * {duj  du)\\LP < C\\E  EtWLp ^ 0,
when t ^  o o ,
for the total mass of the measures duj is uniformly bounded. When t is fixed then Et * {dvj — dv) ^ 0 uniformly on every compact set as j ^ oo, because Et is continuous. Hence it follows that E * dvj ^ E ^ dv in L^^^, To prove the corresponding result about the first derivatives we take a smooth approximation to E^ for example E^{x) = E{x){l — xoi^/^)), where xo ^ CQ^CR^) is equal to 1 in a neighborhood of the origin. Then dE^{x)/dxjdE{x)/dxj
= xo{x/6)dE{x)/dxjE(x){djXQ){x/6)/6
^ 0
in L^ as ^ > 0 if p ( l  n) + n > 0. Hence (£^^  E)' * {duj  du)\\LP > 0 then as 5 ^ 0, uniformly in j , and since E^ * {duj diy) ^ 0 in ( 7 ^ for fixed ^ as j ^^ oo, it follows that u'j ^ u' in L^^^. Prom Fatou's lemma it follows that
/
( l i m Uj{x))da{x)
> lim /
Uj{x)da{x).
In view of the general inequality (3.2.6) it will follow that (3.2.6) must be an equality almost everywhere with respect to da if we show that (3.2.8)
/ Uj{x)da{x)
^ / u{x)da{x),
when j ^ oo,
and that the righthand side is finite. In doing so we may assume that suppdcr C y , and then the statement follows from the fact that (E ^ dvj){x) da{x) = = j{E
* da){y)dv,{y)
E{x — ^ J{E
y)dvj{y)da{x)
* da){y)du{y)
 J{E
*
du){x)da{x),
where the limit is justified by the continuity of £^ * da. To prove that Ujda ^ uda it suffices to show that we may replace da by i/;da in (3.2.8) if 0 < 0 G Co, and that follows if we prove that E * (ipda) is continuous. Now the fact that J5t * dcr  J5 * da as t ^> — oo implies by Dini's theorem that the convergence is locally uniform, that is, {Et — E) ^ da —^ 0 locally uniformly as t ^  o o . Now 0 < {Et  £") * ipda < {snpip){Et — JS) * dcr, so it follows that also E * ipda is continuous. The proof is complete.
152
III. S U B H A R M O N I C F U N C T I O N S
When E^ da is continuous, it follows that a subsequence of the sequence Uj converges to u almost everywhere with respect to a. However, this is not true for the full sequence. To give an example we choose for z/ = 1, 2 , . . . a finite set Ajy C {y E H^; \y\ < v] such that min^^^^ E{x — y) < —v^ when a; < z/. Ordering the functions £^( — y)lv with 7/ G Aj, as a sequence Uj^ we have Uj —> 0 in Ll^^i^^) but lim ^ ^ "^jC^) — ~c>o for every x. The condition on da in Theorem 3.2.13 is obviously fulfilled if da = 0 dx where 0 < I/J G CQ^ and dx is the Lebesgue measure, so lim^^.oo'^j = '^ almost everywhere with respect to the Lebesgue measure. We may also replace the Lebesgue measure by the area measure dS in any C^ hypersurface, for the area of its intersection with a ball of radius e is 0{e'^~^)^ which suffices to show that {Et — E) t (^dS) —> 0 locally uniformly as ^ ^ —oo. We leave the verification as an exercise but shall give some comments on the case of a hyperplane. P r o p o s i t i o n 3.2.14. Let X be an open set in H^, K a compact set in R'^'^j and / a compact interval on R such that K x I C X. Then there exists a positive constant C such that for all subharmonic functions u in X (3.2.9) /
\u{x',Xn)
U{x\yn)\dx'
'(X), so if 0 < x ^ CQ^{X) then the mass of du = xA?x has a uniform bound. Choose x so that X = 1 in an open set Y D K x I. As in the proof of Theorem 3.2.13 the function v ~ u — E ^ dv is harmonic in Y and we have uniform bounds for V and its derivatives in K x I. Now (3.2.11)
/
\E{x',Xn)
 E{x',yn)\dx'
 \\\xn\  bn,
for it suffices to prove (3.2.11) when 0 < 2/n < ^n? with all absolute value signs removed, and then it follows from the fact that the integral of 2dE{x\xn)ldxn = PH{X.,0) with respect to x' is equal to 1. Hence we obtain / \{E^du){x\xn)
 {E^diy){x',yn)\dx'
<   x n  yn\ / du,
which proves (3.2.9). Since / JK
\u{x',Xn)\dx'
< / JK
\u{x',yn)\dx'
\C\Xnyn\
\u\dx, JX
yn ^ I,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
153
the estimate (3.2.10) follows when we integrate with respect to yn over / . For the restriction of subharmonic functions to hyperplanes we also have a Holder type continuity: P r o p o s i t i o n 3.2.15. Let X be an open set in W^, K a compact set in R ^  ^ and assume that K x {0} CX. Then (3.2.12) / \vi{x\0)V2{x',0)\dx' u{Ox) is a subharmonic function (see Exercise 3.2.1), it follows that x y^ M(0, \x\) is subharmonic. This is equivalent to the convexity stated in the theorem. It is of course also a consequence of (3.2.13), and since dfi can be any positive measure we see from (3.2.13) that one cannot improve on the convexity statement. However, when we study plurisubharmonic functions in Chapter IV, then dfi cannot be arbitrary, and Mu{x,r) will have a stronger convexity property. Exercise 3.2.4. Let X be an open set in R'^, a;o ^ X^ and let IA be a subharmonic function in X \ {XQ}. Show that M ( x o , r ) is bounded above as r —> 0 if and only if for some R> 0 one can write u = v \ w where v is subharmonic when \x — XQ\ < R and w is harmonic when 0 < x — xo < R with M>uj{xojr) = 0 when 0 < r < R. (Hint: Conclude that Mu{xQ^r) is increasing and that the mass of Au near XQ is finite.) Show that u can be extended to a subharmonic function it in X if and only if u is bounded above in a neighborhood of XQ, and that u{xo) is then the limit of M{xo^r) as r ^> 0. Conclude that in Exercise 3.2.1 one can allow any analytic map (p : X ^ X. Exercise 3.2.5. Prove that ME{x^r) Msix^r) = e(r) if r > \x\.
= E{x) if r < x and that
T h e o r e m 3.2.17. Let u be a subharmonic function in R^ such that Mu{0,r) = o(logr) as r —> oo. Then u must be a harmonic function. Proof. Since Mi^(0,r) is an increasing convex function of log r, by Theorems 3.2.3 and 3.2.16, it must be constant if it is o(logr) as r —> oo. Thus Mu{0,r) = u{0) for every r, and it follows from (3.2.13) that Au = c6o for some c > 0. Hence u — cE \ h where h is harmonic, and since Mu{^^r) — ce{r) + /i(0) we conclude that c = 0. T h e o r e m 3.2.18. If (p is convex and increasing on R, cp{—oo) = limt_*_oo r, then (p{u{x)) < ip{Mu{x,r))
= cp{
u{x\ry)duj(y)/cn)
J\y\=i
< / J\y\=i
^{u{x^ry))du}{y)/cn
=
M^i^^){x,r),
BASIC FACTS ON SUBHARMONIC FUNCTIONS
157
where the first inequaUty follows from Theorem 3.2.3 (ii) and the fact that if is increasing, and the second follows from the convexity of (p and Jensen's inequality (see Theorem 1.1.14 and Exercise 1.1.9). This proves the theorem, for it is clear that ^{u) is upper semicontinuous. Remark. It suffices to assume that ^p is defined in an interval containing the range of IA, for ip can then be extended to the whole line with preservation of monotonicity and convexity. Exercise 3.2.6. Let l i i , . . . ,Uk be subharmonic functions in X C R^, let / be a convex function in an open convex set in R^ containing the range of ('Ui,... ,1^^), and assume that / ( s i , . . . ,Sfc) < / ( t i , . . . ,tfc) if both sides are defined and Si < ^ i , . . . , ^ ^ < tk Prove that f{ui^,..^Uk) is subharmonic. Deduce that log(e^^ + • • • + e^^) is subharmonic and that {u\ \ •' • {vF^Yl'P IS subharmonic if p > 1 and all Uj are nonnegative. Corollary 3.2.19. U u is subharmonic when Ri < \x\ < R2 and (p is convex and increasing on R, with (p{—oo) = limt_,oo ^{t)j then M^p(^u){0^ r) is a convex function of e{r) when Ri < r < R2; ifu is subharmonic when \x\ < R then M(p(it)(0,r) is a convex increasing function of e(r) when 0 < r < R with limit ip{u{0)) as r —> 0. In the following results we just discuss subharmonic functions in an annulus and omit the obvious improvement in the case of a ball. T h e o r e m 3.2.20. Ifu is subharmonic log ( /
when Ri < \x\ < R2 then
e 00 it converges to max^=i '^{^y)^ which proves the statement. Remark. We can also prove the theorem directly and more easily from the definition of subharmonic functions. If i?i < r i < r < 7*2 < i?2 and u{x) < Mj when \x\ = rj, then u{x) < aE{x) + b when r i < \x\ < r2 provided that ae(rj)\ b = Mj. Thus maxa.=^ u{x) < ae{r) fb then, which proves the statement. In the same way we also see that im.x^x\=r'^i^) is increasing when 0 < r < i? if TX is subharmonic when \x\ < R.
160
III. SUBHARMONIC FUNCTIONS
Corollary 3.2.23 (Hadamard's three circle t h e o r e m ) . If f is an analytic function in the unit disc in C, then (3.2.14)
(ij/(,e'«)Prf0)'
is a logarithmically convex increasing function of log r G [—00,0), if 0 < p < oo. (We interpret (3.2.14) as max^ /(^e*^) ifp — oo.j T h e o r e m 3.2.24. Ifu is a subharmonic function in all of R^ and u{x) < o(log \x\) as X ^ oo, then u is a constant. Proof. By Corollary 3.2.22 we know that M{r) — max^=^tA(x) is a convex increasing function of logr with limit ?x(0) as logr —> —oo, and by hypothesis it is o(logr") as logr ^ +oo. Hence it must be constant, so u{x) < u{0) for every x. We can move the origin to any other point so it follows that 16 is a constant. Corollary 3.2.25 (Liouville's t h e o r e m ) . If f is an entire analytic function in C sucii that \f{z)\ = o{\z\) as z ^^ oo, then f is a constant. Proof. By Theorem 3.2.24 we know that log \f{z)\ is a constant. If / ^ 0 we conclude that log f{z) is locally an analytic function with constant real part, hence constant imaginary part, so f{z) is a constant locally, hence globally. Corollary 3.2.23 is obviously false in general when ^ < 0. However, a closely related result is true in an important special case: T h e o r e m 3.2.26. If f is an analytic function in the unit disc in C, /(O) = 0, and / is injective (^^schlicht^^) then (3.2.14) is a logarithmically concave increasing function of logr G (—oo,0) ifp < 0. Proof
By Theorem 3.2.20 we know that f27r
l0g(
/"\p'°8l/(re)rfg/27r)
is a convex function of logr when 0 < r < 1, for plog/(>2:) is harmonic when 0 < z < 1. What must be proved is that it is decreasing, that is, that
I
/o
Jo
is an increasing function of r. Since r—\og\f{re'')\^Re{dlogf{z)/d\ogz)
=
lmid\ogf{re'')/de),
BASIC FACTS ON SUBHARMONIC FUNCTIONS
161
we obtain with the notation / = i?e^^ that the derivative is \p\I{r)/r I{r)=
[ ^ Jo
where
R{re'ydip{re^^)/dede.
We must prove that / is positive. The argument (p oi f is uniquely defined for 0 < ^ < 27r when it is chosen for ^ == 0, and we have /^ ^ dcp/dO dO = 27r. Since dip/86 is an analytic function of 6 and not identically zero it has only finitely many zeros for fixed r, corresponding to a finite number of critical values for ip. Let A be an interval C [0, 27r] containing no critical values mod 27r. Then the equation (^(r, 0) = a mod 27r has a fixed number of simple zeros 6i,... ,6^ which are C^ functions oi a E A, corresponding to Ri < R2 <   < RkThese represent the intersections of the Jordan curve [0,27r] 3 0 i> f{re^^) with the ray where argi^; = a. The interior Gr — {f{z)\ \z\ < r} contains re*^ when r < Ri but not when i?i < r < i?2, and so on, so it follows that k is odd since the last intersection must lead to the exterior. Since / preserves the orientation the set Gr must lie to the left of the oriented boundary curve, which proves that {—iy~^dip/d6 > 0 at 6j, so I{r) can be written as a sum of integrals / {Ri{af  R2{af f • • • I Rk{af) JA Since Ri{aY  R2{a)P > 0, . . . , Rk2{ocY  Rki{aY is positive, which proves the theorem.
da. > 0, the integrand
Theorem 3.2.26, due to Prawitz [1], has as consequences some important classical results of Koebe and Bieberbach: T h e o r e m 3.2.27. Iff is as in Theorem 3.2.26 then \f"{0)\ the range of f contains {w G C; \w\ <  / ' ( 0 )  / 4 } , and
r^9i,^
J ^ M _ < l/'WI < J_±l£L
< 4/'(0),
ui/i
Proof. We can assume that /'(O) = 1. By Theorem 3.2.26 w i t h p =  2 a , a > 0, the integral r27r /•ZTT
J{r)=
/ Jo
\f{re^')\'^de/27r
is decreasing. To compute J{r) we note that since f{z)/z has no zeros in the simply connected unit disc, we can define g{z) = {f{z)/z)~^ uniquely as an analytic function with ^(0) = 1. Then we have 27r
nZTT
J{r)^
/
id\i2^2a \g{re"')\'r"'dd/2'K.
162
III. SUBHARMONIC FUNCTIONS
li g{z) — XI0^ ^i^"^ ^^ ^he Taylor expansion then CQ = 1 and oo
oo
J{r) = ^ r  2 0  )  c , f ,
rJ\T)
0
^Y^r''^^^^{13
 2a)c,f < 0.
0
If a < 1 then all terms with j > 0 are positive and we obtain when r* —> 1 oo
1
S i n c e / ( z ) / z = l + 2 / " ( 0 ) / 2 + O(^2) we have ^(z) = which means that c\ = —a/"(0)/2, so we have
\azf'{^)j1^0{z^),
o(la)/"(0)/22  / ' ( 0 )  / 4 . If Id < 1 then
h{z)=^fi{z0/iCzl))f{0 satisfies the hypotheses in the theorem, and since
(zl we have Hz) = / ' ( O d C P  l){z + z'O + / " ( O d d '  l ) ' ^ V 2 + and the inequahty \h"{0)\ < 4/i'(0) gives
/"(C) /'(O
C77
2CP 1^ 4d < ild^liKP
Oiz%
BASIC FACTS ON SUBHARMONIC FUNCTIONS
163
Prom the proof of Theorem 3.2.26 we know that Re{zf'{z)/f'{z)) rdlog\f'\/dr, so we obtain
=
2r4 4 + 2r ^ < 9 1 o g  / \/dr < —. 1 — r^ 1 — r^ Integration from r = 0 gives (3.2.15) and completes the proof. Remark. Bieberbach conjectured that \f''^\0)/n\\ < n  / ' ( 0 )  for every n, which he proved when n = 2. The conjecture was finally proved by De Branges [1]. There and in all results in Theorem 3.2.27 there is equality for the Koebe function f{z) = z/{l + z)^ mapping the unit disc to C slit from I to +00 along R. Already in Exercise 3.2.1 we observed that subharmonicity is invariant under orthogonal transformations. Convexity, on the other hand, is invariant under arbitrary linear transformations, and this is the essential distinction between the two notions: T h e o r e m 3.2.28. Let u be defined in an open set X C R " , and assume that UA{X) = u{Ax) is subharmonic in XA = {x;Ax G X} for every nonsingular linear transformation A. Then u is locally convex. Conversely, if u is a locally convex function in X then UA is locally convex, hence subharmonic. Proof. Let / C X be a closed interval on a line, with distance > ^ > 0 to dX. For the sake of simplicity we assume that / is on the a;iaxis. By hypothesis U£{x) = u{xi^ex2^ • • •, sxn) is then subharmonic at distance < 6 from / for every e G (0,1). Hence u{x) — Ue{x)
(iii) we start from a function UQ with the properties listed in Theorem 3.2.30. By adding a constant we can attain that lio < 0 in K. Let M = {x e X \ Y;uo(x) < 1}. This is a compact set. It follows from (ii) that for every x £ M we can find a harmonic function h in X such that h < 0 in K and h{x) > 1. Then we have /i > 1 in a neighborhood of x. By the BorelLebesgue lemma we can therefore find finitely many functions / i i , . . . , h^ which are harmonic in X and < 0 in if, such that maxj hj > 1 in M . But then it follows that u{x) — m3.x{uo{x),hi{x)^... ^hN{x)) has the properties required in condition (iii), which completes the proof. In the whole section we have emphasized that l o g  /  is subharmonic if / is an analytic function in an open subset of C. Equivalently, log \u'\ is subharmonic if ix is a harmonic function and \u'\ is the Euclidean norm of u'. We give as an exercise to prove an analogue in R^:
BASIC FACTS ON SUBHARMONIC FUNCTIONS
167
Exercise 3.2.7. Let uhe a, harmonic function in an open set X C R"^, n > 3, and set \u'\ = {J2^ \du/dxj\'^)'2. Prove that \U'\'P is subharmonic if P^ (n —2)/(n — 1) but that this is not always true when j9 < (n —2)/(n —1). (Hint: Calculate A(?z'p + e)^^ when e > 0 and examine the result when the matrix u" is diagonal, with trace 0.) Finally, in analogy with Section 1.7, we shall discuss when the minimum of a family of functions is subharmonic. As a preparation we first prove an analogue of Theorem 1.7.3. T h e o r e m 3.2.32. If X C R^ is an open set, I = [a, 6] is a compact interval on R, and u G (7^(X x / ) , then U{x) = min^^/ u{xj t) is in C^'^{X) if and only if (i) u'^{x^t) does not depend on t when t G J{x) — {t £ I]u{x^t) = U{x)] and x e X is fixed. (ii) For every compact K C X there is a constant AK > 0 such that, with dj = d/dxj, n
(3.2.17)
Y^ \djdtu{x, t)\'^ < AKd^u{x, t),
ifx e K,t e J{x) \ dl.
1
Then U E C^ in the open subset YofX Y = {x eX]t^dI For every x eV (3.2.18)
deGned by
and dlu{x,t)^Q
when t e
one can find t e J{x) C I\dl
such that
djdkU{x)
= djdku{x,t)

J{x)}.
{djdtu{x,t)){dkdtu{x,t))/d^u{x,t),
when j,k = 1 , . . . , n . For almost all x £ X the equation (3.2.18) is valid for every t G J{x) \ dl with d^u^x, t) > 0, and (3.2.19)
djdkU{x)
= djdku{x,t),
for every t G J{x) with t e dl or d^u{x,t)
j,k =
l,...,n,
= 0.
Proof. That (i) is necessary and sufficient for U to be in C^ follows as in the proof of Lemma 1.7.2. If A is a Lipschitz constant for U' in a convex compact set K C X, then it follows from the proof of Theorem 1.7.3, applied to {s, t) \^ u{y f sv^ t) with y £ K and a unit vector v^ that {{da:,v)dtu)^ < AKdlu{x,t), AK = A{
sup xeK,tei,\v\=i
lixeK.te
J