The set of extreme points of a compact convex set By G(inAN BJ/SRCK

(1)

A R K I V F O R M A T E M A T I K B a n d 3 n r 42 Communicated 5 J u n e 1957 by OTTO FI%OSTMAI~

T h e set o f e x t r e m e p o i n t s o f a c o m p a c t c o n v e x set

B y G(inAN BJ/SRCK

0.1. Introduction

The purpose of this note is to establish necessary a n d sufficient conditions for a set S in a Fr~ehet space X to be the set of extreme points of some compact convex set. 1

We shall first s t u d y the case where X is a Euclidean space R n. Suppose t h a t S is the set of extreme points of some compact convex set K. As is well known, S need not be closed. B u t it is also well known t h a t K must be the convex hull of S and, consequently,

the closure o/ S is contained in the convex hull o/ S. (b) F u r t h e r obvious necessary properties of S are t h a t

the closure o/ S is compact (a)

and t h a t

no point o/ S is a barycenter o/ a /inite set o/ other points o/ S. (c') The following is obviously an equivalent formulation of (c'): F o r all A c S, no point of S is in the convex hull of A without being in A, or, with obvious notation,

S n H ( A ) ~ A /or all A c S. (e)

Thus, in a Euclidean space, conditions (a, b, c) are necessary. I n Theorem 2.1 t h e y are proved to be sufficient. We shall also prove t h a t some apparently stronger conditions are necessary.

When X is a general Frgchet space (Theorem 3.1), we have to make some changes in conditions (b) a n d (e), for it develops t h a t (b) is not necessary a n d (a, b, c) are n o t sufficient. However, if the convex hull occurring in (b) a n d (c) is replaced b y a certahl larger "hull", the resulting conditions will be necessary and sufficient. This new hull H e of a set S was considered b y Choquet [4, 5, 6]. I t is defined as the set of barycentra of those positive R a d o n measures /z of total mass one on ~q which are contained in S in the sense t h a t (<q-S) has #-measure zero. Choquet [5 or 6] proved t h a t each compact convex set K in a Fr@chet space is the He-hull of its set of extreme points. The Krein-

1 F o r t e r m i n o l o g y , see [1] a n d [2].

(2)

GORA_N BJORCK, The s e t of extreme points of a compact c o n v e x s e t

Milman theorem only tells us t h a t every point of K is the b a r y c e n t e r of a measure on ~q. Choquet's result means t h a t in a certain sense there need n o t be a n y masses outside of S. A t first it might seem more natural to require the masses to be contained in S in the stronger sense t h a t each x in K is the barycenter of a measure with its support (compact b y definition) contained in S.

However, it follows from examples given by Choquet (see L e m m a 3.2), t h a t this is n o t possible. Accordingly we shall f~nd t h a t the corresponding " h u l l ope- ration" (the Hst of section 0.2) cannot be introduced in conditions (b) a n d (c).

0.2. Notation and definitions

Let X be a linear space. L e t S c X and let K c X be convex. We shall consider the following sets:

m m

Hm ,es, a,_>0, Ea, = 1};

o o

H (S) = convex hull of S = 0 Hm (S);

m - - 1

and E ( K ) = the set of extreme points of K

= { x e g l x e H 2 ( A ) ~ x e A for all A c K}.

When X is a topological linear space, we shall also consider:

S = c l o s u r e of S;

(S) = H (S) = closed convex hull of S ; H c (S) = (x E X ] 3 R a d o n measure/z on

w i t h x = I t d p ( t ) , #>_0, # ( S ) = 1, p ( S - S ) = 0 } ; and

Hst (S), defined as H c (S) b u t with " # ( S - S) = 0" replaced by " s u p p o r t (#) c S " . (The support is the smallest compact set outside which # vanishes.)

Evidently,

S = H1 (S) ~ Ha (S) ~ . . . ~ H (S) c Hst (S) c Hc (8) ~ H (S).

If x a n d ~u are related as in the definition of Hc (S), we shall write x =

f

^{t d ~ .}

(z)

1. Non-topological preliminaries

I n this section, K (convex) a n d S are sets in a linear space X.

We first formulate without proof two remarks which are simple consequences of the definition of an extreme point:

1.1. Remark. I f E (K1) c K s c K1, then E (K1) c E (Ks).

1.2. Remark. E (H (S)) c S for all S.

1.3. Lemma. I] S = E (K) /or some convex K , then S = E (H (S)).

Proof: Take K I = K and K 2 = H ( S ) in R e m a r k 1.1. We get S ~ E ( H ( S ) ) , and the lemma follows from R e m a r k 1.2.

(3)

ARKIV FOR MATEMATIK. Bd 3 nr 42 1.4. Remark. I n contrast with the case in Theorem 2.1, it does not follow from S = E (K) t h a t K = H (S).

1.5. Lemma. The following three properties o / a point x and a set S are equivalent:

(I) ~ ~ E (H (S)).

(II) x E H ( A ) implies x E A /or all A c S . (III) x E H (B) implies x E B /or all B ~ H (S).

This was observed b y Klee [7, Theorem 1]. (That ( I ) ~ (III) is v e r y near the definition of an extreme point, ( I I I ) ~ (II) is obvious, and ( I I ) ~ (I) is proved b y straightforward calculation.)

The following lemma states t h a t condition (e) of section 0.1 is necessary and sufficient for S to be the set of extreme points of some convex set.

1.6. Lemma. The/ollowing three properties o / a set S are equivalent:

(1) S = E ( K ) [or some convex K.

(2) SN H ( A ) c A / o r all A c S . (3) S N H (B) c B / o r all B ~ H (S).

Proof: Obviously, (3) implies (2). N e x t suppose (2) is true and let x E S . Then x satisfies (II) of L e m m a 1.5, and from (I) we get S ~ E ( H ( S ) ) . R e m a r k 1.2 t h e n completes the proof of (1) with K = H ( S ) . Finally, suppose (1)is true and let x E S N H ( B ) with B c H ( S ) . F r o m L e m m a 1.3, x satisfies (I) of L e m m a 1.5, and so from (III) we see t h a t x EB, which completes t h e proof of (3) and of L e m m a 1.6.

2. C o m p a c t c o n v e x s e t s in a E u c l i d e a n s p a c e

2.1. Theorem. The/ollowing three properties o / a set S in R ~ (n > 1) are equivalent:

(1) There is a compact convex set K c R n, such that S = E ( K ) . (2) (a) ~q is compact,

(b) S ~ H ( S ) , and

(e) S N H ( A ) ~ A /or all A c S . (3) (a) S is compact,

(b) S ~ H n _ I ( S ) , and

(c) S N H (B) ~ B /or all B c H (S).

Proof: E v i d e n t l y (3)~ (2), so t h a t we only have to prove t h a t ( 2 ) ~ ( 1 ) ~ (3).

First, suppose (2) is true. F r o m (2a), H(~q) is compact [1, exc. 2]. B u t from (2 b), H(~q) c H (S), and so H(~q) = H (S). Further, from (2 c), L e m m a 1.6 and L e m m a 1.3, we get S = E ( H ( S ) ) . Thus (1) is proved with K = H ( S ) .

Finally, suppose (1) is true. Then K = H ( S ) = H ( S ) , (see e.g. [1, exc. 9]). (3a) is evident and (3 c) follows from L e m m a 1.6. Since H (S) is closed, we get Lqc H (S), or b y Carath~odory's theorem, ~ c H n + I ( S ) . I f S is closed, (3b) is evident. If not, let x E ~ q - S , and let L be a supporting plane of H ( S ) through x. Since x E H (S) N L = H (S N L), we get from Carath6odory's theorem t h a t x E Hn (S N L)

c H n ( S ) . First, if S c L , we replace / ~ b y L, repeat the argument and find x EHn_l (S). Next, if S contains points outside of L, let a be such a point, and let L a g a and Lb be the open halfspaces produced b y L. L e t b be a n y point E Lb. Suppose x ~ H=-I (S). Then x must be an interior point of a simplex T i n

(4)

GORAN BJORCK, The set of extreme points of a compact convex s e t

L with its vertices in S ~ L. L e t O be the interior of H ( T U {a} U {b}). Since 0 is a neighborhood of x, there is in 0 a point y e S = L ~ U L. Then y is an interior point of H ( T U {a}) or of T. This violates condition (3 c) with B = {a} U {vertices of T}, a n d the proof of (3 b) a n d of T h e o r e m 2.1 i s complete.

2.2. Corollary. F r o m (3 b), we get the well-known result t h a t if K is a com- p a c t convex set in R ~, t h e n E(K) is closed.

2.3. R e m a r k . I n T h e o r e m 2.1, condition (3 b) cannot be replaced b y ~q= Hn-~ (S).

Indeed, we shall construct a set S, satisfying e.g. condition (1) b u t n o t t h e proposed condition. (For n= 3, the same example is given in [1, ext. 8].) We consider R ~ as Rn-2× R 2 (with the coordinate spaces imbedded). I n R ~-2 we t a k e a simplex W w i t h the origin o as an interior point. L e t V be t h e set of vertices of W. T h e n o ¢ H , - 2 (V). The two remaining dimensions will assist in m a k i n g o an accumulation point of e x t r e m e points. I n R *, let C be t h e circumference of a circle t h r o u g h o. Finally, take S = V U ( C - {o}). Then, since o e H (V)c H (S), we find H ( S ) = H (V U C). B u t V U C is compact, and so H(S) is compact, a n d E ( H ( S ) ) ~ V U C. On the other hand, a point x e C - { o } o b v i o u s l y cannot be t h e barycenter of masses which are p a r t l y distributed in W. Hence, since x E E (C), we get x e E ( H ( S ) ) , and similarly for the points of V. Thus S = E ( H ( S ) ) . Fin- ally, we see t h a t o ¢ H~_~ (S).

3. C o m p a c t c o n v e x sets i n a F r ~ c h e t space

3.1. Theorem. The /ollowing three properties o/ a set S in a real Frdchet space X (i.e. a complete, metrizable, locally convex, real vector space, e.g. a real Banach space) are equivalent:

(1) There is a compact convex set K c X, such that S = E (K).

(2) (a) ~ is compact, (b) ~ q = H c ( S ) , and

(c) S A H c ( A ) c A /or all A ~ S . (3) ( a ) = ( 2 a )

(b) = (2 b)

(c) S A H c ( B ) ~ B ]or all B ~ H ( S ) .

Proof: As in T h e o r e m 2.1, we shall prove t h a t ( 1 ) ~ (3) and ( 2 ) ~ (1). Suppose (1) is true. T h e n K = H c ( S ) = H ( S ) [5, th6or~me 1, or 6, th6or~me 1]. Then (3b) follows f r o m S c H c ( S ) = H c ( S ) , a n d (3a) is obvious. To p r o v e (3c), let B c K a n d let x E S ( I H e ( B ) . Hence x = f t d t z . B u t xEF_,(K), a n d so # m u s t con-

(B)

sist of one u n i t y pointmass, which m u s t then be in B. Consequently, x E B, which completes t h e proof of (3).

Finally, suppose (2) is true. Since S is c o m p a c t and X is complete, H(~q)is c o m p a c t [1, N ° 1]. F r o m ~ q ~ H ( S ) we get E r ( S ) = H ( ~ q ) , _ a n d so H_(S) is compact. N e x t we shall prove t h a t E ( H ( S ) ) c S . L e t x E E ( H ( S ) ) = E ( H ( S ) ) . T h e n x Ekq [1, prop. 4], and b y (2b), x EHc(S). As in the first p a r t of the proof, f r o m x = f t d # it follows t h a t x E S. We h a v e thus proved the inclusion E ( H (S)) = S,

(S)

a n d we shall n o w use this to prove the reversed inclusion. I n (2 c), we take A = E ( H ( S ) ) . F r o m Choquet's theorem, H e ( A ) = H ( S ) . Hence f r o m (2 c), we get S ~ A , which completes t h e proof of (1) a n d of t h e o r e m 3.1.

(5)

ARKIY FOR MATEMATIK. Bd 3 nr 42 I n sections 3.3-3.5 we shall discuss the possibilities of m a k i n g certain changes in the conditions of T h e o r e m 3.1. W e shall need the following lemma, which states t h a t in general Hst (E (K)) :4: Hv (E (K)).

3.2. L e m m a . There exist in some Erdchet space Y a compact convex set W and a point a E W such that a ~ H s t ( E ( W ) ) .

I n fact, t h e following e x a m p l e is given b y Chequer [6, p. 17]: L e t U (with elements u) be the closed interval [0, 1] and let V (with elements v) be the set of reals modulo I. T h e n Y is t h e space of R a d o n measures m on U × V a n d W is the set of m E Y such t h a t m >_ 0, m (U × V) = 1 a n d dm (u,v) =dm (u,v + u).

L e t a E W be the homogeneous distribution of t o t a l mass one on U × V. W e find t h a t E (W) is the set of m E W satisfying:

(1) the s u p p o r t of m is contained in the circle u=um, where Um is a con- s t a n t (depending on m);

(2) if u~ is irrational, t h e n m is t h e homogeneous distribution of t o t a l mass one on the circie u =um ; a n d

(3) if um =p/q, where p a n d q > 0 are relatively prime integers, t h e n m con- sists of q s y m m e t r i c a l l y distributed point-masses, each with mass 1]q.

Then it is easily seen 1 t h a t with a certain # with the s u p p o r t of # n o t contained in E ( W ) , we have a=Stdl.t. F r o m a uniqueness t h e o r e m [4, thdor~me

(E(W))

1, or 6, th6or~me 2), it follows t h a t a ~ H ~ ( E ( W ) ) .

3.3. R e m a r k . I n T h e o r e m 3.1, condition (3 b) cannot be replaced b y ~qhHst (S).

I n fact, let Y , W a n d a be as in L e m m a 3.2. I f we t a k e X = Y × R ~ a n d pro- ceed in a way similar to t h a t used in section 2.3 (with o replaced b y a), we find t h a t S = E ( W ) U ( C - { a } ) satisfies condition (1) of T h e o r e m 3.1 a n d t h a t

~¢H8~(S).

3.4. Remark. I n T h e o r e m 3.1, condition (3 c) m a y be replaced b y t h e a p p a - r e n t l y weaker condition:

(3 cl) S N H 2 (B) c B /or all B c Hc (S).

I n fact, (3 cl) is exactly the s t a t e m e n t S c E ( H c (S)). To prove t h a t (3 a, b, cl) implies (1), we observe t h a t we have as in the second p a r t of the proof of T h e o r e m 3.1, t h a t E ( H ( S ) ) c S . I f we take the He-hull of b o t h m e m b e r s of the last inclusion, we get H ( S ) ~ H c ( S ) . Consequently these two sets are equal, which completes the proof t h a t S = E ( H ( S ) ) .

3.5 Remark. I n contrast with R e m a r k 3.4, condition (2 c) of T h e o r e m 3.1 cannot be weakened to

(2Cl) Sfl H s t ( A ) c A /or all A ~ S .

We consider t h e following example. L e t X = Y and S = {a) U E (W) with Y, W, a n d a as in L e m m a 3.2. Clearly S satisfies conditions (2a, b) b u t n o t (1) of T h e o r e m 3.1. We shall now prove t h a t S satisfies (2cl). L e t A ~ S and x ES N Hst(A). First, suppose x = a . I f a E A , there is nothing to prove. I f a ~ A , it follows t h a t A ~ S - { a } , and hence x q H ~ t ( S - { a } ) , or in t h e n o t a t i o n of L e m m a 3.2, aEH~t(E(W)), which is n o t true. Finally, suppose x # a . Hence

x e ( S - { a } ) N H~t(A)= E (W) N H ~ t ( A ) ~ E ( W ) N H c ( A ) ~ A , by (3 e) of T h e o r e m 3.1.

I F o r t h e c o n c e p t of i n t e g r a l of a f a m i l y of m e a s u r e s , see [3, § 3, N ° 1].

(6)

CORAN BJORCK, The set of extreme points o f a compact convex set

R E F E R E N C E S

1. B O U R ~ I , N . , E s p a c e s v o c t o r i e l s t o p o l o g i q u e s , c h a p . I I , § 4. A c t u a l i t 6 s Sci. I n d . 1189, P a r i s 1953.

2. - - I n t 6 g r a t i o n , c h a p . I - - I V . A c t u a l i t 6 s Sci. I n d . 1175, P a r i s 1952.

3. - - I n t 6 g r a t i o n , c h a p . V. A c t u a l i t 6 s Sci. I n d . 1244, P a r i s 1956.

4. CHOQU'ET, O., U n i c i ~ d e s r e p r 6 s e n t a t i o n s i n t 6 g r a l e s a u m o y e n do p o i n t s e x t r 6 m a u x d a n s los c 6 n e s c o n v e x e s r6ticul&s. C . R . A c a d . Sci. P a r i s 243, 5 5 5 - 5 5 7 (1956).

5. - - E x i s t e n c e d e s r e p r 6 s e n t a t i o n s i n t 6 g r a l c s a u m o y e n d e s p o i n t s e x t r 6 m a u x d a n s les c 6 n e s c o n v e x e s . G.R. A c a d . Sci. P a r i s 243, 6 9 9 - 7 0 2 (1956).

6. - - E x i s t e n c e e t u n i c i t 6 d e s r e p r 6 s e n t a t i o n s i n t 6 g r a l o s a u m o y c n d e s p o i n t s e x t r 6 m a u x d a n s les c 6 n e s c o n v c x e s . S 6 m i n a i r e B o u r b a k i , d6c. 1956.

7. K L E % V. L., A n o t e o n e x t r e m e p o i n t s . A m e r i c a n M a t h . M o n t h l y 62, 3 0 - 3 2 (1955).

T r y c k t d e n 15 oktobor 1957

Uppsala 1957. Almqvist & Wiksells Boktryckeri AB