Israel Journal of Mathematics, Volume 1, Issue 1 (1963)

ON A CONJECTURE OF LINDENSTRAUSS* BY VICTOR KLEE ABSTRACT It is proved that each n-dimensional centrally symmetric convex polyhedron admits a 2-dimensional central section having at least 2n vertices. Some other related results are obtained and some unsolved problems are mentioned.

The following conjecture of Joram Lindenstrauss was transmitted to me by Branko Griinbaum: (E) I f P is an n-dimensional centrally symmetric convex polyhedron, then some 2-dimensional central section of P has at least 2n vertices. I present here a proof of this conjecture as well as a dual statement (I) about affine images of polyhedra. In fact, the simplest approach to (Z) seems to be by way of (I), using the duality between sections and affine images as in [2, 3]. Let us begin with the proof of (I), where of course (Z) and (I) both require that n>2. (I) If P is an n-dimensional centrally symmetric convex polyhedron, then some 2-dimensional affine image of P has at least 2n vertices. Proof. The assertion is obvious for n = 2. Suppose it is known for n = k and consider the case n = k + 1, where we assume without loss of generality that P lies in a (k + 1)-dimensional real vector space E and is centered at the origin 0 of E. We wish to show that some 2-dimensional linear image of P has at least 2k + 2 vertices. Let V be the set of all vertices of P and let ~ be a linear transformation of E onto a k-dimensional vector space F such that ~ is biunique on V. Then ~P is a k-dimensional centrally symmetric convex polyhedron, so (by the inductive hypothesis) there exists a linear transformation ~/' of F onto a 2-dimensional vector space G such that the convex polygon .Q' = q'~P has at least 2k vertices. Let ~ ( F , G) (resp. ~ 0 ( F , G)) denote the space of all linear transformations of F into G (resp. onto G); the spaces .~(E, G) and ~e0(E , G) are similarly defined. In the following paragraphs, it is convenient to topologize the spaces E, F, and G by means of Euclidean metrics, the spaces -~¢(F, G) and -~(E, G) by means of uniform norms, and the space ~ of all convex polygons in G by means of the Received December 3, 1962 * Research supported in part by the National Science Foundation, U.S.A. (NSF-GP-378).

1

2

VICTOR KLEE

[March

Hausdorff metric. However, at the cost of some extra effort it would be possible to dispense with these assumptions and carry out the proof for vector spaces over an arbitrary ordered field. Noting that the function ~,~P] ~ • ~e(F, G) maps £P(F, G) continuously into ~ , that the function (number of vertices of K) ] K • ~ is lower semicontinuous on g(', and that the set J = (~k•Lo(F,G):~b is biunique on ~V} is a dense open subset of ~ ( F , G), we see the possibility of choosing q in J close to ~/' such that the convex polygon Q = ~/¢P has at least 2k vertices. When Q has at least 2k + 2 vertices, there is no problem. When Q has only 2k vertices, we shall produce in G another linear image Q + of P which has more vertices than Q. Production of Q+ depends on the following simple fact: (?) I f Q is a convex polygon in the plane G, A is the set of all vertices of Q, and B is a finite subset of Q ,.~ A, then there exists e > 0 such that whenever B + is a finite set for which B + eg Q but B + lies in the e-neighborhood of B, then the polygon Q÷ = conv(A d B +) has more vertices than Q has. Here it is essential that the set B should not include any vertex of Q, and this accounts for our interest in transformations of E which are biunique on the set V of vertices of P. Suppose Q has only the 2k vertices ---ql .... , +-qk. For each i there is a unique vertex Pi of P such that r/¢pi = qi. Let H be a k-dimensional linear subspace of E such that {Pl . . . . ,Pk} C H and let ¢ denote the restriction of t/~ to H. Then of course ~(P 0 H) = Q. Let u • P ~ H, so that each point x of E admits a unique expression in the form x = x' + x"u with x' • H and x" • R (real numbers). For each point z • G, let the transformation ~b, e ~ o ( E , G) be defined as follows: 4~,(x' + x"u) = ~x' + x"z.

Note that with Zo = r/~u, we have ¢~o(X' + x"u) = ~x' + x"rl~u = nCx' + x"~lCu = rl~(x' + x"u), so ¢,o=n~. Let Go denote the set of all z e G for which the transformation ~bz is biunique on V. Then z o • Go, and we claim further that the set G ,-, G O is finite. Indeed, if z e G ~ Go there exist v • V and w • V such that v • w but ~b,v = ~b:w, whence ~(v' - w') = (w" - v")z. If v" = w", then ~v' = ~w' and r/~v = r/~w, contradicting the choice of r/. If v"=/=w", then z = ( w " - v " ) - l ~ ( v ' - w'). Thus there are only finitely many possibilities for z e G ~ Go. Let zl ~ Go ~ Q. Since Zo e Go and the set G ~ Go is finite, there exists zl/2 • Go such that the segments [z o, z~/2] and [z~/2, zl] both lie entirely in Go. Define z~ = ( 1 - 2 2 ) z o + ( 2 2 ) z l / 2

for 0 < 2 < 1 / 2 ,

zx = ( 2 - 2 2 ) z x / 2 + ( 2 2 - 1 ) z x

for 1 / 2 < 2 < 1 .

Then the function Note that

~,aPl,~ • [0,1]

and

is a continuous mapping of [0,1] into ~ .

1963]

ON A CONJECTURE OF LINDENSTRAUSS ~bzaP D ( ( H A P ) = Q

3

for all 2 ~ [0, 1], and that

~bzoP = ~/~P = Q

while

u ~ P and ¢~1u = zl(~Q.

Let p be the least upper bound of those 2 e [0,1] for which CzxP = Q. Then # < 1 and ¢z~P = Q, but there are values of 2 arbitrarily close to # for which Q is properly contained in the polygon ¢,aP. Of course CzxP = conv¢,~V for all 2 e [0, 1], and since zu e Go we have ~bz,(V "~ { + P l ..... +- Pk}) = Q "~ {+- ql ..... +- qk}Applying the italicized statement (t) above, we see the existence of 2 close t o / t for which the convex polygon Q + = tk~xQ has more vertices than Q and hence, being centrally symmetric, has at least 2k + 2 vertices. The proof is now completed by mathematical induction.| It seems worthwhile to provide a more metric form for (I). (H) I f P is an n-dimensional centrally symmetric convex polyhedron in E', there is a 2-dimensional plane T in E n such that the orthogonal projection z~ of E" onto T carries P onto a convex polygon uP having at least 2n vertices. Proof. By (I) there is a linear transformation z of E" onto /~2 such that zP has at least 2n vertices. Let T 1 = z-1(0) and let T be the orthogonal supplement of T 1 in E'. | Proof of (Z). Let P be an n-dimensional centrally symmetric convex polyhedron, centered at the origin of an n-dimensional real vector space E. Let ( , ) be an inner product on E and let pO be the polar body { y ~ E : ( x , y ) < 1 for all x ~ P}. Then of course p0 is an n-dimensional centrally symmetric convex polyhedron, and by (H) there exists a two-dimensional linear subspace T of E such that the convex polygon ztP° has at least 2n vertices, where u is the orthogonal projection of E onto T. Since TA(PNT) °=T~clconv(P°UT

O) = T N ( P

° + T O) =

T N ( riP° + TO) = ~ze°, it follows that the polygon P 0 T has at least 2n vertices. (The relevant basic results on the polarity o may be found in [1] and in [ 2 ] . ) | In the absence of symmetry assumptions, the methods used for (H) and (~) lead to similar results in which the number 2n is replaced by n + 1. In the nonsymmetric version of (E), there exist 2-dimensional sections with at least n + 1 vertices through each interior point of the n-dimensional convex polyhedron P. There remain many interesting problems concerning the numbers of faces of sections or projections of convex polyhedra. Some of these may be formulated as

4

VICTOR KLEE

follows. Suppose ~ is an indexed family ((P,, X,): t e I}, where, for each t ~ I, P, is an n-dimensional convex polyhedron in E" and X, is a nonempty subset of E". For 0 ~ j < k = an-1 > ... > a[(n+2)/21

or

(2.4)

an ~ a l > a n _ 1 ~ a 2 ~ . . . ~

a[(n+l)/2]

holds. The set (a) is strictly symmetrically increasing if (2.5)

al = an >=a 2

=

an-1 > ... > (=)a[(n+ 2)/z].

For a given set (p) = (Pl ..... Pn) there exist, in general, two distinct, symmetrically decreasing rearrangements. The rearrangement ordered as in (2.1) is denoted by ( p - ) = (p;-, ...,p~) so that (2.1')

Pl- < P~ < P ; < P~--1 < ... --< Pt~n+2)/21 ;

the other symmetrically decreasing rearrangement is denoted by ( - P ) ----( - P l , . . . , - P,) :

(2.2')

-Pn =< -Pl -- + p1>=+ p n--1 => + p2>-~

> +P[(n+l)/2]"

"'" =

Finally, if (p) admits a strictly symmetrically increasing rearrangement, i.e. if (p +) = (+p), then this rearrangement is denoted by (p*) = (p*, ..., p*): (2.5')

=

-----

= P n* - t

>---- "'" ~ ( --- ) P [ ( n* + 2 ) / 2 ] "

Sets (p) admitting such a rearrangement (p*) are called paired. For even n a set (p) is paired if every value occurs an even number of times; for odd n the smallest

14

BINYAMIN SCHWARZ

[March

value has to occur an odd number of times, every other value an even number of times. If (q) = (ql ..... q,) is a rearrangement of (p) = (Pl ..... p,) then we say that the diagonal matrix Q = (ql ..... q,} is a rearrangement of the diagonal matrix P = {Pl, ..., P,}. Given P, its rearrangements P - , - P , P+, +P and, if P is paired, P* are defined by the order relations (2.1')-(2.5') for their elements. Together with these diagonal matrices of order n, we consider also Jacobi matrices of the same order for which the elements in the diagional are equal to the real constant k and the elements in the super- and sub-diagonal are - 1. (All other elements are 0). We use the notation (cf. (1.4)) k -1 (2.6)

-1 k

-1

A~" ) = Ak = -1

(Aktx) = ( k ) , A~k2 ) =

k -1

-1 k

..). For the corresponding quadratic forms

1

the following two lemmas hold: LEMMA 1.

Let n--I

(2.7)

A~k")(y,y) = Ak(y,y ) = k ~ y2 _ 2 ~ YiY~+I /=I

i=i

be the quadratic form belonging to the matrix (2.6). (i) I f k > 2cos(Tr / n + 1) then Ak(y , y) is positive definite; (ii) if k < 2 cos (it / n + 1) then Ak(y , y) takes negative values; (iii) if k = 2cos(~r/n + 1) then Ak(y,y ) is non-negative definite. This lemma is an immediate consequence of the following theorem due to Fan, Taussky and Todd [2, Theorem 9]. I f yl,..., y, are n real numbers, then l1

•

i=0

2

7~

]~

2

( Y , - Yi+ 1)2 > 4sin -~n-+-])',=o y'

(where Yo = Y.+x = O) unless Yi = c~, where (2.8)

Yi= s i n - - - - n+l'

i = 1,.

""

n.

LEMMA 2. Let Ak(y, y ) be defined by (2.7) and let the set (y) of n non-neoative numbers be given except in arrangement. Then Ak(y , y) attains its minimum if

1963]

BOUNDS FOR FREQUENCIES

15

(y) is arranged in symmetrically decreasing order. Moreover, if all the elements of (y) are positive and if no three elements of (y) have the same value, then Ak(Y, Y) attains its minimum only if(y) is symmetrically decreasing. In our notation, this can be stated as (2.9)

Ak(y, y) > Ak(y-, y- ),

y, > O, i = 1,..., n,

with the additional statement that, if Yi > 0, i = 1, ..., n and if no three numbers Yi are equal, then equality holds in (2.9) only if (y) = ( y - ) or (y) = (- y). We remark that ~i=xYi n 2 is invariant for all rearrangements of a given set. Defining

S(y,y) : ~ y2+ ~ YiY,+I,

(2.10)

i=1

i=I

it follows that (2.9) is equivalent to

S(y,y) < S ( y - , y - ) .

(2.11)

This last inequality is a very special ease of a Theorem of Hardy, Littlewood and P61ya on bilinear forms. [5, Theorem 371; to obtain (2.11) set, in their notation, c o = 1, c 1 = c_ 1 = ½, all other c = 0; let their two sets (x) and (y) coincide and if (our) n is even, let one element of (y) be zero.] This proves the first part of Lemma 2. As mentioned, both parts of the lemma follow from Lehman's result. Indeed, if we apply the corollary of [7] to the funetionf(x) = x 2 we obtain that n-1

~ YiYi+l,

i=1

i=l,..'n,

yi>=O,

is maximum if (y) is symmetrically decreasing and that under the more restrictive assumptions on (y) this maximum is attained only if (y) is symmetrically decreasing. This is clearly equivalent to Lemma 2.

3. The minimum of the least characteristic value. THEOREM 1. Let Ark")= A k be the Jacobi matrix of order n defined by (2.6) and let Q = {ql .... ,q,), qi > O, i = 1. . . . ,n be any diagonal matrix of order n with positive elements. Denote the least characteristic value of the pencil A k - 2Q by 2t(Q). Let P = {Pl .... ,P,}, P i > O, i = 1.... ,n be a given diagonal matrix and let P - and P + be its symmetrically decreasing and increasing rearrangement respectively.

if (i)

k > 2 cos

7[

n '+ 1'

16

B I N Y A M I N SCHWARZ

[March

then (3.1)

2x(P) _>_2~(P-).

Moreover, if 7~

(i')

2 > k > 2cos----~, =

n-t-

then equality holds in (3.1) only if P itself is symmetrically decreasing.

if 7g

(ii)

k < 2 cos - - - - n+l'

then (3.2)

2~(P) > 21(P + ).

Finally, if 7~

(iii)

k = 2 cos ~ n+l'

then (3.3)

2a(Q) = 0

for any diagonal matrix Q = {ql . . . . . qn}, qi > O, i = 1..... n. Proof. For given n and k let y = (Yl ..... Yn) be the characteristic (principal) column vector corresponding to the least characteristic value 2~(P) of the pencil A k - - 2P[3, p. 310]. Let lyl = (I yt [ .... ,1 y,[), then n-1 i=l

[2

n

Ak(y,y)= k ~ y ~ - 2 ]E y i Y i + l i=l

>=k

Ely /

n-1

-2]E

i=l

i=1

lyillyi+xl= Ak(lyl, lyl),

and

P(y,y)= ~ piy~= ~ i=l

i=l

p, Iy, I

=P(lYI,lYl).

It now follows from the minimum characterization of 2x(P) [3, p. 319] that [y[ is also a characteristic vector corresponding to 2t(P). But the recursive form of the n scalar equations of (3.4)

Aky = 21(P)Py

shows that the characteristic vector belonging to 21(P) is determined to within a scalar factor. We may therefore assume y = and it follows from (3.4) that y > 0, i.e. y~ > 0 for all i, i = 1..... n. ['Cf. 4, p. 136].

l yl

1963]

17

BOUNDS FOR FREQUENCIES

Assume now that (i) holds. By Lemma 1, Ak(y, y) is in this case positive definite, hence 21(P) > 0. Let y be the positive characteristic vector belonging to 21(P) Then

(3.5)

Ak(x,x) = 21(p ) = Ak(y,y__)_ > A k ( Y - , Y - ) • > min P-(x,x'-----~) P(Y, Y) = P - ( Y - , Y - ) = :,,o

0).

Here y - = (YT, ..., Y~) is the symmetrically decreasing rearrangement of y: (3.6)

(0 ~ f(IYF

(6)

i=1

i=1

-y7+~]). (y,.+l=Yl,

Yn'-+l= Yl),

where ( y - ) = (y~,...,y~)is the symmetrically decreasing arrangement of (y) satisfying (1'). To prove the second assertion of the theorem we have to show that equality holds in (6) only if (y) is of circular symmetry. The proof proceeds by induction. For n = 2 and n = 3 every set is of circular symmetry and (5) is clearly invariant under all rearrangements, hence the theorem holds trivially for those n. We now assume the validity of the theorem for sets of n - 1 numbers and show that this implies its validity for sets of n numbers. Let (y) = (Yl,"', Y,) be such a set. Without loss of generality we may assume that (7)

0 = Yt 0).

(7) and (9) imply

~-,f(ly,-y,+xl)=

(11)

i=1 n-1

= ~ : f ( I x , - x , + l I) i=1

i=1

+f(xl) + f ( ~ - l ) -f(Ixl-xn-i [),(y.+l= y , - - 0 ; x~=xO.

Similarly, it follows from (8) and (10) that n-I

~f(ly7

i=I

- y~l])

= Z f(lxl - xi+~ l) +f(x;) +f(x~_ ~)-f(lx~

(y;+~= We define

- x'_t

i=I

y/- -- o; x,', = xD.

l),

A RESULT ON REARRANGEMENTS

1963]

s>0,

g(s,t) = f ( s ) + f(t) - J ( l s - t I);

(13)

25 t>0.

(11)-(13) give

Z f ( I y , - y,+ll)- i ~= 1 f ( l y ; - y-+~t)

i=1

(14) n-1

= Z f(lx,-

n-I

x,+~ 1) -

i=1

Ef(lx"

-

x'+~ l)

g(xl,x,,_O - g(xl,x'-O.

+

i=l

(7)-(10) imply that (x') is a rearrangement of(x). (1') and (10) give (15)

X n' - 1


1) (Sk = al + a2 + ... + ak) be a real or complex sequence satisfying k a k = O(1), and define (2.11)

fl(x)=e -e-~,

7(x)= {01

x - log n

(2.12) f ( x ) = k = l a k

+a,+llog(n

+ l)_logn,

x~0Xc~> 0 Proof of

(2.25)

If(x) - f ( y ) ] < K~.[ x - y I,

K being dependent only on 6. Thus the lim defining L* in (2.22) certainly exists. In order to prove our Theorem it is enough to show that (2.21) holds; the fact that A* is the best possible constant satisfying (2.21) follows from the remark after Theorem 2. Let now be e > 0 given; define 6 > 0, x o > 0 such, that for x > Xo, Y _>-Xo, Ix - Y] => 6 (by (2.22)) (2.26)

If(x) - f ( y ) [ < (L* + e) [ x - y l,

and for x > Xo, y > Xo and Ix - y[ < 6 (since f(log x) is slowly oscillating) (2.27)

If(x) - - f ( y ) [ < s.

Now f(tl) - Tp(y) =

(f(tl) - f ( x ) ) d ( 1 - fl(y - x)) oo

(2.28) =

Let y , q > x o + 6. By (2.25)

=0

E r; + axo

(~/-y)(1-fl(y

Jq-,~

Xo))+

f)

+~= It + I 2 + I 3 + 1 4

udfl(u) --XO

and by (2.5) and (2.6) we obtain easily (2.29)

I t = o(1)

as y -~ oo.

36

A. MEIR

By (2.26) 112 [ < (L* + e

( t / - xld(1 - fl(y - x l ) < (L* + e o

dfl(u vy--TI

dt ,J y - T i

(2.30) = (L* + e)"

(1 - fl(u))du

I n the same way (2.31)

]14] _-
0. Thus X has property C*(qS) with ~b(2) = 21/p and (6.2) is equivalent to (2.7). On the other hand, taking V = B*(p) we have, by (1.3) and (6.1),

M,,(x)

=

(ll x II/p)''.

Therefore oo

(6.3)

II x,,lt 1,p < n=l

is a necessary and sufficient condition for the absolute convergence of (1.1). It follows that in every complete infinite-dimensional space with a p-homogeneous norm there exist series which are unconditionally yet not absolutely convergent (take e.g. 7, = n -p (n = 1,2 .... )).

1963]

ON SERIES IN LINEAR TOPOLOGICAL SPACES

45

The theorem now follows from the fact that the topology of a locally bounded space can always be given by a p-homogeneous norm with a suitable 0 < p < 1. (S. Rolewicz [7], see e.g. I-6] p. 165). 7. Proof of Theorem 4. It may be assumed without loss of generality that the topology is given by a translation invariant metric. Let Yl be an arbitrary non-zero point of the space, and having determined y~ chose Y~+I # 0 so that (7.1)

1

A(yi+a) < TIIY'II,

(i =

1,2 .... ).

Given any sequence of numbers ~ ( i = 1, 2 .... ), it follows from (7.1) that the series

~, (~Yi

(7.2)

i=1

is metrically, hence absolutely, convergent. Therefore, by completeness, it represents a point in the space. Let Y be the totality of points representable by series (7.2). It is obviously a linear set and the representation (7.2) isunique. Indeed, we claim that if (7.2) has the value zero then all (~ vanish. Assume (~ = 0 for i < j, (j > 1); if (g # 0 then

1 ~,(,yi=yj+

~,

(,

contradicting (7.1). Thus ~ = 0 (i = 1,2 .... ). Moreover, Y is a closed subspace. Indeed, let oo

(7.3)

z, = •

~,,~Yi (n = 1,2,...)

i=1

be a sequence of points of Y converging (in the original space) to z. The z, form then a Cauchy sequence and, as in the proof uniqueness above, it follows that each sequence ~1,i, (2 ~..... ~,,,~... is a Cauchy sequence. Hence limn = co~,,~ exists (i = 1,2 .... ). If we denote this limit by ~ then it follows immediately from (7.1) that the sequence (7.3) tends to the point represented by (7.2). Hence z z Y and Y is closed, therefore also complete. We have in fact shown that a sequence (7.3) is convergent if and only if all the sequences (1,i, (2,i .... ,(.,i .... (i = 1,2 .... ) are convergent. Therefore, a series (1.1) with cO

(7.4)

x, = Z

~,,~Yi

i---1

is unconditionally convergent if and only if all the series

46

ARYEH DVORETZKY

[Ma rc h

oo

(7.5)

~: ~. i,

(i = 1,2 .... )

tl=l

are unconditionally, hence absolutely, convergent. But if the series (7.5) are absolutely convergent it follows that the Minkowski functional (1.3) with

v -- {x: IIx 11--- IIy J Ill satisfies co n=l

oo

j

~ I ~.,, I!1 Y, II < ~"

Mv(x.) S ~ n=l

i=l

II 11-~

Since Yi 0 by (7.1) it follows that the unconditional convergence of (1.1) implies its absolute convergence and the theorem is established. 8. Proof of Theorem 5. Since X is metrizable we can define its topology by a translation invariant metric d'. We now introduce another metric d, equivalent to d', having the required properties. Let g(t) be strictly increasing and continuous for 0 < t < ~ with g(0)= 0 and such that oo

(8.1)

~ ?.g(~,) < ~ . n=l

Put f(t)=tg(t). Then f(t) is continuous and strictly increasing from 0 to ~ and we have for 0 < s, t < s t

g(s + t)> max(g(s),g(t)) > - - - g ( s ) =

=

+ ----g(t),

s+t

s+t

or

(8.2)

f(s + t) >_f(s) + f(t) ;

and (8.2) obviously remains valid when either s or t, or both, vanish. L e t f -1 be the inverse function o f f and define

a(x, y) = f - l ( a ' ( x ,

y))

for all x, y ~ X. Then d(x, y) is again translation invariant and induces the same topology as d'(x,y). But IIx. II --=v(S), for all coalitions S, S c B ~ b. If b is a c.s. and S is a coalition we denote by P(S, b) the set of partners of S in the c.s. b, i.e. the set U{B : B eb, #(S ca B) > 0}. DEFINITION 1.3. Let (x, b) be c.r.p.c, and K and Ldisjoint, non-null coalitions with the same partners. An objection of K against L in (x, b) is a c.r.p.c. (y,c) that satisfies: #(P(K,c) ca L) = O, y(t) > x(t) for almost every t e K and y(t) > x(t) for almost every t e P(K,c). DEFINITION 1.4. Let (x, b) be a c.r.p.c, and (y, c) an objection of a K against an L in (x, b). A counter objection of L against K is a c.r.p.c. (z,d) that satisfies: I~(K - P(L, d)) > O, z(t) > x(t) for almost every t ~ P(L, d), and z(t) > y(t) for almost every t E P(L, d) ca P(K, c). A c.r.p.c. (x,b) is stable if every objection in (x, b) can be countered. The bargaining set Mo is the set of all stable c.r.p.c.'s. The following two lemmas are not difficult to prove. LE~tA 1.5. Let u(t) be a real measurable function on a coalition S. If 0 < 0 < #(S) then there is a coalition T c S such that /~(T) = 0 and inf{u(t): t E T} __>sup {u(t): t ~ S-T}. LEMMA 1.6. Let (x,b) be a c.r.p.c., K = {t:w(t)> x(t)} and L = {t:x(t)> max(0, w(t))}. If a coalition K1 c K has an objection (y,c) against a coalition S such that f l~(w - y)d# + f ~ ( w - x)dl~ < f S~L(X -- w)d#, where K2 = {t :t ~ P(K 1, c), w(t) > y(t)} and K 3 = K - P(K1, c),then S has no counter objection.

2. Stable payoff configurations. THEOREM 2.1. Let (m,w) be an m-quota game, ba m.e.s, and (x,b) a e.r.p.c. If x(t) < max(O, w(t)) a.e. then (x, b) e M o.

50

BEZALEL PELEG

[March

Proof. We have to show that if (y, c) is an objection of a coalition K against a coalition L, then L has a counter objection. We denote A = {t:w(t) 0 let U e c be an m-coalition; we assert that there is a sub-coalition U1 c U such that p ( U 1 ) = m - r , # ( K - Ux) > 0 and .fv~(W - y)d# > O. I f y(t) = w(t) for almost every t e U, then we have to delete f r o m U a sub-coalition U: c U whose measure is r and that satisfies g(U 2 ~ K) > (r/m)l~(U n K) to obtain U 1. I f S = {t:t e U, y(t) > w(t)} has positive measure, let U2 be a sub-coalition o f S that satisfies 0 < # ( U 2 ) < r/2. Denote V 1 = U - U2. f v , ( W - y ) d # > O . I f # ( K - 111) = 0 let further U 3 be a subcoalition o f 1/1 n K that satisfies 0O, where 1"2 = Vt - U3./t(K - 112) > 0. So we can always obtain a coalition V c U such that fv(w - y)dl~ > O, I~(V) > m - r and #(K - V) > 0. Let U 1 be a subcoalition of V such that # ( U t ) = m - r and inf {w(t) - y(t) : t e U1} > sup {w(t) - y(t) :t e V - Ut}. U1 has the desired properties. To complete a counter objection of L when r > 0 , La can f o r m an m-coalition F together with U1. The payments to the members of F will be w(t) for t e L 3 and y(t) + (1/m - r) ~vl(w - y)dp, for t e UI. A consequence o f Theorem 2.1 is COROLLARY 2.2. Let (re, w) be an m-quota game and b a m.c.s.; there is always a measurable function x such that the c.r.p.c. (x, b ) e M o . THEOREM 2.3. Let (m,w) be an m-quota game and b a m.c.s, that satisfies s(b) ~ 2. If a c.r.p.c. ( x , b ) e M o then x(t) < max(O,w(t)) a.e.,

Proof. 1 = mq + r, 0 < r < m. W.l.o.g. b = { B 1, ...,Bq, Bq+i}, #(B~+I) = r and # ( A n (Bq_ 1 U Ba)) = 0, where A = {t :w(t) < 0}. We denote also A1 = A - Bq+ i and R = B q + I - A . Let (x,b) be a c.r.p.c. We denote K = { t : t e B j , j < q , x(t) < w(t)}, J = {t : t e B i, j < q, w(t) = x(t)} and L = {t :x(t)>max(O,w(t))}. We shall prove that the inequality #(L) > 0 implies that (x, b) ~ M0. This will be done by proving the existence o f sets U and V such that U has an objection against 11, and V has no counter objection. We distinguish the following possibilities:

(a)

#(K) + #(J) + #(R) < m

Let T c L UA1 be a coalition whose measure is m - / x ( K ) - #(R) - #(J) and t h a t satisfies inf{w(t) - x ( t ) : r e T} ~ sup{w(t) - x ( t ) : t e ( L U A I ) - T}. P((L u A i ) - T,b) = P(K,b). We have J'xur(W - x)d# > 0. Since s(b) > 2 we

1963]

QUOTA GAMES WITH A CONTINUUM OF PLAYERS

51

have also S L - T ( x - - w ) d # > J ' T ( X - w)d#. K can object against (L u A 1 ) - T by forming, together with R u J u T, an m-coalition F; the payments to the members of F will be w(t) for t ~ R U J, x(t) for t e T and

z(t) = x(t) +

(w(t) - x(t)) fK~T(W -- x)d# J"K(W -- X) d.u

for t e K . Since f r ( W - z)d# = S K ( w - x ) d # by l e m m a 1.6 (L u A 1 ) -

(b)

fK ~ r( w -- x)d# = ~ r ( X - w)d# < ft.- r ( X - w)d#

T has no counter objection.

#(K) + p(R) < m and p(K) + #(R) + p(d) > m

K can object against L U A 1 by forming, together with R and a sub-coalition of J whose measure is m - # ( R ) - #(K), an m-coalition with a quota split. L UA1 has no counter objection.

(0

a(K) + t,(R) >_-m

p(K)+p(R)=pm+s, O<s<m. I f s = 0 K objects against L U A 1 by forming, together with R, p m-coalitions with a quota split. I f s + p(J) ~ m let Q = K be a coalition whose measure is s. K can object against L u A1 by letting K - Q f o r m together with R p m-coalitions with a quota split, and Q form an additional m-coalition with a quota split together with a sub-coalition of d whose measure is m - s. In b o t h cases L U A1 cannot counter object. So we may assume in the following that s > 0 and s + p(J) < m. We now show that we m a y also assume that #(L) > m -- s - #(J). I f #(L) < m - s - #(J) let Q c K be a coalition whose measure is s that satisfies sup {w(t) - x(t) : t ~ Q} < i n f {w(O - x(t) : t e K - Q}. B~ c P ( K - Q, b). K - Q can object against G = P ( K - Q, b) - K - J by forming, together with R, p m-coalitions with a quota split. . [ a , ~ L ( x - - w ) d # > ~ [ B ~ n z ( X - - w ) d # = f n q , ~ r ( w - x)d# > S Q ( w - x)d#, so by l e m m a 1.6 G cannot counter object. Let now S be a sub-coalition of K whose measure is s that satisfies sup {w(t) - x(t) : t e S} < inf{w(t) - x(t) : t ~ K - S}, and T a sub-coalition of L U A l whose measure is m - s - #(J) that satisfies i n f { w ( t ) - x(t) : t e T} ~_ sup {w(t) - x(t) :t ~ (L U AI) - T}. We distinguish the following sub-cases:

(c.1)

fs (w-x)d~O vT

52

BEZALEL PELEG

[March

In this case there is a coalition S 1 c K such that/ffS1) = s , P ( K - St, b)D B~uBq_ 1 and f s , ( W - x ) d l ~ < I L ~ B q u S q - , ) ( X -- w)dlt. I f P(K - S,b) ~ Bq WBq_I we may choose S 1 = S. If P ( K - S, b) gp Bq _ 1 we may assume that Bq_ 1 n K = S. Let 0 < 6 < min (s,l~(Bq n K ) ) and let U 1 c S and U 2 c K n B q be coalitions whose measure is 6. Sa = (S - Ua) W U2 has the desired properties. Now we can construct an objection of K - Sx against G = P(K - S~, b) - K - J by letting K - S x form together with R p m-coalitions with a quota split. By lemma 1.6 G has no counter objection.

f s u r (w - x)dlt > 0

(c.2)

We have that P(K, b) = P((L u A 1) - T, b) ; also f r.(x - w ) d g > 2 f T(X -- w)dlt since f T ( X - - w ) d # < IBjoL( x - w ) d # for j = q - - l , q . K has the following objection against (L w At) - T : K - S forms, together with R, p m-coalitions with a quota split, and S joins J u T to form an additional m-coalition F ;the payments to the members of F will be x(t) for t e J u T and

z(t) = x(t) +

(w(t)- x(O) Is u r ( w - x)d~ I s (w - x)d#

for t ~ S . Since IL_T(X--W) d # > J ' r ( x cannot counter object.

w)dkt = I s ( W -

z)d#(L uAt) - T

COROLLARY 2.4. Let (re, w) be an m-quota game and b a m.c.s. I f w(t)>O a.e. then a c.r.p.c. (x, b) ~ M o if and only if x(t) < w(t) a.e. To complete the p r o o f of corollary 2.4 we have to show that if s(b) = 1 and (x, b) ~ Mo then x(t) < w(t) a.e. ; we omit the details. This work was carried out under the supervision of Dr. R. J. A u m a n n as a part of a doctoral thesis to be submitted at the Hebrew University.

REFERENCES 1. Aumann, R.J., July 1962, Markets with a continuum of traders I. Econometric Research Program, Research Memorandum No. 39. Princeton University. 2. Aumann, R.J. and Maschler, M., The bargaining set for cooperative games. To appear in No. 52 o f Annals of Mathematics Studies, Princeton University Press, Princeton, N.J. 3. Davis, M. 1961, Symmetric solutions to symmertic games with a continuum of players, Proceedings of the Princeton University conference on recent advances in game theory, held in October 1961. 4. Kalish, G.K., 1959, Generalized quota solutions of n-person games, Annals of Mathematics Studies No. 40, Princeton N.J. pp. 163-177.

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QUOTA GAMES WITH A CONTINUUM OF PLAYERS

53

5. Maschler, M., December 1961, Stable payoff configurations for quota games, Econometric Research Program, Research Memorandum No. 36, Princeton University. 6. Peleg, B. On the bargaining set M0 of m-quota games, To appear in No. 52 of Annals of Mathematics Studies, Princeton University press, Princeton N.J. 7. Shapley, L.S. and Milnor, J.W., Feburary 1961, Values of large games, II: Oceanic Games, The Rand Corp., RM-2649, THE HEBREW UNIVERSITY OF JERUSALEM

ON O R D E R PRESERVING CONTRACTIONS* BY S. R. F O G U E L ABSTRACT

Let (f~,Y.,~) be a measure space and let P be an operator on L2(f~,Y.,p) with [IPI]< 1, Pf~ 0 a.e. whenever f > 0. If the subspace K is defined by

K--{x I

I[pnxll=ltP*nxll:-Ilxll

,

n--1,2,...)

then K = L2 (f~,~l,/~), where [1 c E and on K the operator P is "essentially" a measure preserving transformation. Thus the eigenvalues of P of modulus one, form a group under multiplication. This last result was proved by Rota for finite # here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. Introduction. The purpose o f this note is to generalize the results o f [2]. In [2] R o t a studies the eigenvalues o f modulus one o f a contraction P on L 1 (S, ~ , / 0 where # is a finite measure and P satisfies the following:

a. Pf > 0 w h e n e v e r f > 0. b. ess. sup. I Pf[ Z ess. sup. If[" This problem is related to the Frobenius Perron Theory. F o r bibliography on the subject we refer to [2]. O u r generalization is two-fold: 1. The measure # is n o t assumed to be finite. 2. The o p e r a t o r P is a contraction on L2(S,E,#) and is n o t assumed to be defined on LI(S, E, p). IfPhasnormonein t and Lo~ then, by the Riesz Convexity T h e o r e m it has n o r m 1 also over L2, thus 2 is weaker than R o t a ' s assumption. We shall use the m e t h o d o f the p r o o f o f T h e o r e m 2.2 and L e m m a 1.2 o f [1]. There the case #(S) < oo was studied. The results o f [2] are included in Theorems 1 and 3 o f this note. Let (S, E, p) be a measure space with/~ > 0. LEMMA 1. Let L be a closed subspace of L2(S,~,,l~), which satisfies: Received March 28, 1963. * The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force. 54

ON ORDER PRESERVING CONTRACTIONS

55

(1) l f f e L then R e f E L. (2) If f is real and f E L then Ifl eL. (3) If f > 0 a . e . , f E L , and c is a positive constant then min (f,c)E L.

Let E' contain all the sets, in E, whose characteristic functions are in L; then (a) The sets in Z' form afield; (b) The characteristic functions of sets in ~,' span L. Proof. Let f, g be real valued functions in L. Then max(f,g) =

k(lf- gl

+f+

g) e Z

min(f,g) = ½(f+g-[f-gl)EL. If tr and z are in E', let I(a) and I(z) denote their characteristic functions. Then max (I(tr), I(z))EL and min (I(tr), I(r))eL or: a U z E E ' and ~ n z E X ' . In order to prove (b) it is enough to show that the only function in L orthogonal to Z' is the zero function. Now if r e L is orthogonal to all functions I(tr),tr e E', then so is Ref. Thus we may assume that./is real. Let f + = ½( lfl + f ) E L and let c be a positive constant. Then, by (3), m i n ( f + , c ) e L and also f + - m i n ( f + , c )

geL. Let ~b = c- i rain (f+, c). Then h, = e- 1 min (5 q~,g) e L. Now: h~(to) = 0 iff÷(~o) < c, since then g(og) = 0, while: he(og) = 1 if f + > c + e. Also, for every to, 0 ~ he(co) ~ 1. Hence he(co) tends to the characteristic function of {~o > c) as ~ --} 0, thus I{co]f+(~o) > c} E Z and is orthogonal to f ; i.e., f < c a.e. for every c > 0. Therefore f + = 0 a.e. Applying the same argument to - f we g e t f = 0.

If+(o,)

REMARK. If p(f~)< OO then 1 E L 2 and Condition (3) is a consequence of Condition 2 and (3')

1 E L.

DEFINITION 1. An operator P on L2(S,X,/t) is called an order preserving contraction (O.P.C.) provided:

1. I f f e L 2 is real valued then so is Pf. 2. I f 0 - < t e L 2 a.e. then Pf>__ 0 a.e. 3. I f f E L 2 is real valued and f < ca.e. then P f < c and P ' f < c.

4. [[P[[ __ 0 and ~ can be chosen arbitrarily small. But then

Rel e-'~PFI > I PFI cos a > I FI

contradicting (*).

(This argument was suggested to us by Y. Katznelson). The proof of the Lemma is now straightforward:

llfll ~ >= II Plf111 [ISlI ~- (PISI,Ifl) >= I(ef,f)l

=

lifll ~

hence (PlSl, ISI~ = II Plsl II IlSll o~ Plsl =IslTHEOREM 1. Let L = {flPf =f} and let Z' contain all the sets tr in ~ such that I(a)¢ L. Then ~' is a field and its characteristic functions generate L, Proof. It is enough to verify Conditions (1), (2) and (3) of Lemma 1.The first condition is obviously satisfied. Now if lYl~ L then ISl ~ L by Lemma 2. Finally if 0 < f = Pf then P[min(f,c)] < Pf =f,

P[min(f,c)] < c ;

thus

P[min(f,c)] < min(f,c). Hence

P [ f - min(f,c)'] = f -

P[min(f,c)] > f -

min (f,c)

a n d f - rain (f, c) => O. We must have equality a.e., since l] P ][ =< 1,thus rain (f, c) e J[. DEF,N,TION

2. K

=

{st II ~sll-- II P*"Sil

:

lifll, n >__ ~}.

1963]

ON ORDER PRESERVING CONTRACTIONS

57

Now [iWf[[ = [[f[[ if and onlyif p , n p ~ f = f , and it is easy to check that P*nP~ is an O.P.C. Also l] P*~fi[ = ][f]] if and only if I~P*~f=f. Thus K is generated by characteristic functions of the intersections of the corresponding subfields of Y~. DEFINITION 3. Let E 1 contain all the sets, a, such that I ( a ) e K. By the above remarks %1 is a field and it generates K. THEOREM 2. The set K is a closed subspace of L2, invariant under P and P*. On K, P is a unitary operator. I f f _LK then weak lim F ' f = weak lim P * f = O.

Also, if tr e E 1, then PI(a) = I(z), where z e %I. Proof. It is enough to prove the last statement since the rest is proved in Theorem 1.1 of [11. Let a e %1 and PI(a) = f then 0 __