SOME CHARACTERIZATIONS OF CONVEX POLYHEDRA
BY
VICTOR KLEE(1) Copenhagen and
Seattle
1. Introduction
This paper deals with subsets of a finite-dimensional euclidean space
E,
a set being calledpolyhedral
or apolyhedron
provided it is the intersection of a finite number of closed halfspaces. Thus as the term is used here, a polyhedron is closed and convex but need not be bounded. A set will be calledboundedly polyhedral
provided its intersection with each bounded polyhedron is polyhedral. Our principal goal is to characterize polyhedra as convex sets, certain of whose projections or sections are polyhedral. I n connection with this task, we are led to develop various properties of polyhedra and of boundedly poly- hedral sets which seem to be available in the literature only for bounded polyhedra or polyhedral cones. Some of our proofs could be simplified a bit b y working in projective space. I~owever, since m a n y of the results on unbounded convex subsets of the atfine space seem to be not so simply obtainable in this way, we have chosen to work entirely in t h e affine space E.Section 2 begins with a simple but useful theorem on the facial structure of an arbitrary convex set, generalizing from the fact t h a t a bounded closed convex set is t h e convex hull of its set of extreme points. This theorem supplies one step in proving equivalence of the following five conditions on a subset K of E: K is the intersection of a finite system of closed halfspaces; K is a closed convex set with only finitely m a n y faces; K is closed and is the convex hull of a finite system of points and rays; K is the closed convex hull of the union of a bounded polyhedron and a polyhedral cone; K is the linear sum of a bounded poly- hedron and a polyhedral cone. Surely the equivalence is generally "known", b u t it seems not to be available elsewhere in precisely this form. I n the present paper, we (1) Based on research supported in part by the Office of Ordnance Research, U.S. Army, in part by a fellowship from the Alfred P. Sloan Foundation, and in part by a National Science Founda- tion Senior Postdoctoral Fellowship (U.S.A.).
8 0 VICTOR KLEE
have need of the equivalence and have included the proof in an effort to smooth the reader's path.
I n w 3 we exploit some connections between polyhedra and convex cones, in prepara- tion for the m a i n results in w 4. These are as follows (rendering "if and only if" b y "iff").
Suppose K is a convex subset of E n and 2 ~<i < n - 1. Then (1) K is polyhedral iff all its i-sections are polyhedral; (2) if K is bounded, K is polyhedral iff all its i-projections are polyhedral; (3) if ?. >/3, K has polyhedral closure iff all its i-projections have polyhedral closure; (4) if K is a cone, K is polyhedral iff all its i-projections are closed. These theorems are believed to be new, except t h a t for ?. = 2 a n d K closed; the result (4) was recently given b y Mirkil [7]. Our general procedure has much in common with his, a n d some of our propositions a m o u n t to formalizations of steps in his proof. The results (1) a n d (2) reduce trivially to the case i = 2, b u t this does not appear to be true of (4).
Section 5 is devoted to some types of "nearly polyhedral" sets, characterizing t h e m in various ways, identifying their polars, etc. The results are applied in discussion of two interesting examples in E a, one a nonpolyhedral closed convex set all of whose 2-dimensional projections are polyhedral, a n d the other a set which is nonpolyhedral even though it a n d its polar are b o t h boundedly polyhedral. I n w 6 it is proved t h a t every convex F , set is a projection of some closed convex set, and the projections of boundedly polyhedral sets are also characterized. Some approximation theorems are given which are valid for un- bounded convex sets and reduce in the bounded case to the classical result on approxima- tion b y po]yhedra.
I n the concluding w 7, the t e r m
polyhedral
is employed in its more customary sense to describe a set which is the union of a finite n u m b e r of geometric simplexes. There is con- structed in E a a nonpolyhedral 3-cell all of whose 2-sections and 2-projections are poly- hedral. Thus convexity seems essential for the results of w 4.F o r basic material on convex sets, including proofs of results used here without specific reference, the reader is referred to [1, 2, 3, 4], especially [4]. I a m indebted to Professor Fenehel for some helpful suggestions, especially in connection with w 2.
Notation and terminology.
A i-//at is a ?'-dimensional affine subspace of E, and the t e r msubsTace
will be reserved hereafter for linear subspaces. Ai-section
of a set X c E is the intersection of X with a i-flat, and a i-projection of X is the image of X under an affine projection of E onto a i-flat. (Since such an affine projection can always be obtained as the composition of a translation with a linear projection onto a i-subspace of E , conditions on the i-projections of a set m a y be regarded equivalently in terms of linear projections.)We denote t h e e m p t y set b y A, and the origin in E b y ~b. Set-theoretic union, inter- section, and difference will be denoted b y U, N, and ~ , the closure, interior, and convex
SOME C H A R A C T E R I Z A T I O N S OF CONVEX P O L Y t I E D R A 81 hull of a set X by cl X, int X, and cony X. The smallest flat containing X will be denoted by fl X and the interior of X relative to fl X by relint X (called the relative interior o] X ) . The relative boundary of X is the set X ~ relint X. For x and y in E, [x, y] = { r x + (1 - r)y:
0~<r~<l}, ] x , y [ = { r x + ( 1 - r ) y : O < r < l } , etc. For X c E , Y c E , s > 0 and A c R (the real number field), Z +_ Y = { x ++ y : x e X , y e Y }, A X = {a x : a e A , x e X }, and S ( X , s) is the union of the open s-neighborhoods of the points of X.
2. Facial structure, polarity, and polyhedra
We prove here a useful result on the facial structure of convex sets, review the notion of polarity, and establish some basic properties of polyhedra which will be used later in the paper. Fenchel's book [4] m a y be mentioned as a basic reference for the methods employed and for Proposition 2.2, Goldman's paper [5] for the result 2.12 (v), and m y paper [6] for 2.3. The remarks on polarity and at least special cases of the results on polyhedra have appeared in print m a n y times (see, for example, [2, 4, 5, 9]).
A convex set will be called reducible provided it is the convex hull of its relative boundary;
otherwise it is irreducible. B y a / a c e of a convex set K we shall mean a convex subset F of K such that whenever x and y axe points of K for which F is intersected b y the open seg- ment Ix, y[, then x E F and y E F. The convex set K is said to be generated by a family :~
of sets provided K is the convex hull of the union of the members of ~. (In using this term, we shall not always distinguish carefully between a point p E E and the corresponding set {p} c E.)
2.1. T~EO~EM. Every convex set is generated by its irreducible ]aces.
Proo]. As always in this paper, we are concerned only with finite-dimensional sets;
2.1 is trivial for those of dimension zero. Suppose it is known for all convex sets of dimension
< n , and consider an n-dimensional convex set K. Let D denote the relative boundary of K. If K ~ cony D, then K itself is an irreducible face of K. If K = cony D, then b y the support theorem K must be generated b y t h e sets H N D, where H is a hyperplane which meets D but not relint K. But each face of a set H N D is easily seen to be a face of K, and by the inductive hypothesis each set H N D is generated by its irreducible faces. This completes the proof.
Theorem 2.1 is especially useful for closed convex sets, in view of the following.
2.2. P R o P o s I T I 0 N. The only irreducible closed convex sets are the fiats and the hall-liars.
Proo]. Obviously all flats and half-flats are irreducible. Now consider an irreducible closed convex set K and let D denote its relative boundary. Since conv D =4= K, we have cony D =~ relint K, and by the separation theorem there must be an open halfspace Q in
6 - - 593804. A c t a mathematica. 102. I m p r i m 4 le 26 s e p t e m b r e 1959
82 VICTOR K L E E
fl K which misses cony D b u t meets relint K. Since K is closed, it can be seen t h a t Q c relint K, a n d is t h e n easy to verify t h a t K = fl K or K is a closed halfspace in fl K.
2.3. COROLnARY. A closed convex set which contains no line is generated by its extreme points and extreme rays.
L e t us include a few more r e m a r k s on facial structure, even t h o u g h t h e y are n o t needed for t h e sequel. A face of a convex set K will be called essential p r o v i d e d it lies in some m e m b e r of each generating f a m i l y of faces of K, a n d strictly essential provided, in addition, it is n o t contained in a n o t h e r essential face. W e shall p r o v e t o g e t h e r t h e follow- ing two results.
2.4. P R O P O S IT ION. Every irreducible/ace is essential, and every strictly essential/ace is irreducible. Thus the strictly essential/aces are exactly the maximal irreducible/aces.
2.5. P R O P O S I T I O N . Every convex set is generated by its/amily o/ strictly essential/aces, but not by any proper sub/amily o/ that.
Proofs. W e note: 1 ~ a convex set is irreducible iff it is an essential face of itself; 2 ~ if X c K a n d F is a face of K, t h e n conv (X N F ) = ( c o n v X ) N F ; 3 ~ if G is a n essential face of F a n d F is a face of K, t h e n G is an essential face of K. T h e assertion 1 ~ is trivial. As for 2 ~ it is obvious t h a t conv (X N F ) ~ (cony X) N F . N o w consider points x~ of X a n d positive n u m b e r s t~ whose sum is 1, such t h a t ~ t~x~ = p E F . T h e n for each j we h a v e p = tjxj + (1 - t j ) y ~ , where y~ = ~ . j ( 1 - t j ) -1 t~x~EK. Since F is a face, it follows t h a t x j E F . T h u s p E conv (X N F) a n d 2 ~ has been proved. Finally, suppose G a n d F are as in 3 ~ a n d ~ is a generating family of faces of K. F r o m 2 ~ it follows t h a t F is g e n e r a t e d b y its family of faces, ( J N F : J E ~ } , a n d t h u s G, being an essential face of F , m u s t be in some set J N F . This establishes 3 ~ , a n d we are r e a d y to prove 2.4 a n d 2.5.
T h a t every irreducible face is essential follows from 1 ~ a n d 3 ~ N o w let $ be t h e family of all strictly essential faces of K. F r o m finite-dimensionality it follows t h a t every essential face (and hence every irreducible face) lies in some strictly essential face, so f r o m 2.1 it follows t h a t $ generates K. Consider a n a r b i t r a r y S E $ a n d let :~ be t h e family of all proper faces of S. I f K is generated b y ($ ~ (S}) U ~, t h e n S, being essential, m u s t lie in some m e m b e r of $ ~ (S} or in some m e m b e r of ~. Since this is impossible, K is n o t g e n e r a t e d b y $ ~ (S} (completing t h e proof of 2.5) a n d S is n o t generated b y ~, whence S is irreducible a n d t h e proof of 2.4 is complete.
I n t h e case of closed convex sets, a more complete picture of t h e facial structure can be obtained f r o m t h e following t w o remarks, whose proof will be left for t h e reader.
2.6. P R O P O S I T I O N . Suppose L and M are supplementary linear subspaces o/ E, J is a convex subset o / M , and xe is the projection o / E onto M whose kernel is L. Then i / G is
SOME C H A R A C T E R I Z A T I O N S O F C O N V E X P O L Y H E D R A 83 a /ace o / t h e set J § L, ~ G is a / a c e o / J , and i / F is a / a c e o / J , xe -1 F is a / a c e o / J + L.
The same assertion is valid/or essential/aces a n d / o r strictly essential/aces.
2.7. P R O P O S I T I O N . Suppose K is a closed convex subset o/ E with r and L = (x : R x c K ) , so that L is a linear subspace o/ E. Let M be a subspace supplementary to L.
Then K = L + ( M N K ) and the closed convex set M N K contains no lines.
I t follows t h a t the essential faces of K are e x a c t l y the sets ~ § L where r is an extreme r a y of M N K a n d t h e sets p + L where p is an extreme point of M N K. These are all strictly essential except for the sets p + L where p is an endpoint of an extreme ray.
F o r a subset X of E, t h e polar X ~ of X is defined as t h e set of all linear functionals / on E such t h a t / x ~< 1 for all x E X . T h u s X ~ is a subset of t h e space E ' conjugate to E, a n d t h e b/polar X ~176 is a subset of E u n d e r t h e usual identification of E with the c o n j u g a t e space of E'.
T h e propositions 2.8-2.10 below summarize some well-known facts which will be used freely in t h e sequel.
2.8. P R O P O S I T I O N . For X c E, the polar X o is a closed convex set which includes the origin ~' o/ E ' , X ~ = [cl cony (X U (~)]o and X 00 = el cony (X U (~b)). T h u s always X ~176176 = X ~ while X ~176 = X i / / X is closed and convex and ~ E X .
2.9. P R O P O S I T I O N . Suppose X is closed and convex with ~ E X . Then X is a ]-subspace i// X ~ is a (dim E - ])-subspace, X is a cone with vertex ~ i// X ~ is a cone with vertex ~', and X is bounded i / / ~ ' E int X ~
2.10. P R O P O S I T I O N . For a / a m i l y ( X a : a E A ) o/subsets o/ E, ( U Xa) ~ = n x ~ and
a e A a ~ A
( fl X~) ~ = el c o n y U X ~
a ~ A a e A
W e shall prove t o g e t h e r 2.11 a n d 2.12 below, supplying five useful characterizations of polyhedra. Essential tools in t h e proof are 2.1, 2.2, a n d parts of 2.8 a n d 2.10.
2.11. T H E 0 R E M. I / K is a closed convex set with ~ E K , then K is polyhedral i// K ~ is polyhedral.
2.12. THEOREM. For a subset K o / E , the following five assertions are equivalent:
(i) K is the intersection o / a / i n i t e system o/ closed hal/spaces;
(ii) K is closed, convex, and has o n l y / i n i t e l y m a n y / a c e s ;
(iii) K is closed and is the convex hull o / a / i n i t e system o / p o i n t s and rays;
(iv) K is the closed convex hull o/the union o / a bounded polyhedron and a polyhedral cone;
(v) K is the linear s u m o / a bounded polyhedron and a polyhedral cone.
Proo/s. W e show first t h a t (i) implies (ii). L e t ~ be a finite family of closed halfspaces whose intersection is K, a n d for each ~ c ~ let V ~ be t h e intersection of t h e bounding
84 VICTOR KSEE
hyperplanes of the members of 6 . There are only finitely m a n y sets of the form K N V 6 , and we shall show t h a t each face of K has t h a t form. Consider an a r b i t r a r y face F of K, and let ~ be the set of all members of ~ whose bounding hyperplanes contain F; obviously F ~ K N V ~ . Now for each G E ~ ~ ~ we m a y choose a point pG of F N i n t G . With p denoting the eentroid of the chosen points Pa, it is evident t h a t p E U = f l i n t G.
Geq~~
Consider an arbitrary point q E K N V ~ . Clearly V| contains the entire line of points Yr = P + r(p - q) (for r E R ) , and since U is open we have y~E U for some t >0. Then y t E K ; since F is a face and pEJq, Yt[, it follows t h a t q E F . Thus K N V| c F and we know t h a t (i) implies (if).
T h a t (if) implies (iii) is a consequence of 2.1 and 2.2, for obviously condition (iii) is satisfied b y every flat and every half-flat.
We next establish
(1) Suppose C e K and K is the closed convex hull of a finite system of points and rays. Then K is the closed convex hull of a finite system of points a n d of rays emanating from r further, the polar K ~ is polyhedral.
L e t ~1 . . . . , Qm and Xm+l .. . . . xn be the given rays and points. For each ~ , let x~ be the endpoint of ~i and let ~ be the r a y ~ - x t , emanating from q~. Since always ~i ~ cl conv ({x,} U e 1) and e~ ~ cl cony ({r U e~), it is evident t h a t g is the closed convex hull of the s y s t e m of points x~(1 ~<i ~<n) and rays el(1 < i < m ) . With y, eeI ~ {r it follows from 2.10 t h a t K ~ is the intersection of n halfspaces of the form { / : / x ~ ~< 1} and m of the form {[ : [y~ <. 0}, so K ~ is polyhedral.
Now consider a closed convex set K 9 r F r o m (1), in conjunction with the fact t h a t 2.12 (i) implies 2.12 (iii), it follows t h a t if K is polyhedral, so is K ~ B u t this implies t h a t if K ~ is polyhedral, so is K ~176 = K, and thus 2.11 has been proved. Now if K satisfies 2.12 (iii), then K ~ is polyhedral b y (1) and K is polyhedral b y 2.11; thus 2.12 (iii) implies 2.12 (i).
We are now in a position to prove
2.13. COROLLARY. A set is a bounded polyhedron i// it is the convex hull o] a / i n i t e set.
2.14. COROLLARY. A set is a polyhedral cone with vertex z i// it is the convex hull o/
a ]inite system o / r a y s e m a n a t i n g / r o m z.
F o r closed sets, these characterizations come immediately from equivalence of condi- tions (i) and (iii) in 2.12. F r o m compactness of [0, 1] it follows readily t h a t the convex hull -of a finite set is compact, hence closed, and this completes the proof of 2.13. F o r 2.14, it
;suffices to show t h a t if x 1 .. . . . x~ are points of E and C is the set of all linear combinations .of the x / s with non-negative coefficients, then C is closed. For/c = 1, this is obvious. Sup-
SOME C H A R A C T E R I Z A T I O N S OF C O N V E X Y O L Y H E D R A 8 5
n - - 1
pose i t is k n o w n for k = n - 1 a n d consider t h e case k = n. L e t J = { ~ t i x ~ : t~ ~> 0}. T h e n
1
C = J + [O,~[x~, a n d J is closed b y t h e i n d u c t i v e h y p o t h e s i s . I f x~EJ, t h e n C = J , so we m a y a s s u m e x ~ J . N o w consider a n a r b i t r a r y p o i n t p E c l C. T h e r e a r e sequences t~ in [ 0 , ~ [ a n d y~ in J such t h a t t ~ - - > t E [ 0 , ~ ] a n d y~ + t~xn--~p. I f t = ~ , t h e n t~ly~ + x~-->r a n d t h u s Xn E J , a c o n t r a d i c t i o n . I f t E [0, ~ [, t h e n ya-->p - tx~, whence p - txn = y E J a n d p E C . T h e p r o o f of 2.14 is c o m p l e t e , a n d we m a y c o n t i n u e w i t h t h e p r o o f of 2.12.
N o t e t h a t in t h e p r o o f of (1) a b o v e we h a v e
m
(*) K = cl c o n y (conv{x~ : 1 ~<i ~<n} U c o n y U 1 ) ,
i=1
so b y u s i n g t h e c h a r a c t e r i z a t i o n s 2.13 a n d 2.14 we see t h a t (iii) i m p l i e s (iv) in 2.12. On t h e o t h e r h a n d , if K is t h e closed c o n v e x h u l l of t h e u n i o n of a b o u n d e d p o l y h e d r o n a n d a p o l y h e d r a l cone w i t h v e r t e x r i t follows b y 2.13 a n d 2.14 t h a t K h a s t h e f o r m (*), b y (1) t h a t K ~ is p o l y h e d r a l , a n d t h e n b y 2.11 t h a t K is p o l y h e d r a l . T h u s ( i v ) i m p l i e s ( i ) i n 2.12, a n d i t r e m a i n s o n l y t o p r o v e t h a t (iv) a n d (v) a r e e q u i v a l e n t . To t h i s end, we e s t a b l i s h (2) S u p p o s e X is a c o m p a c t c o n v e x set w i t h C E X a n d Y is a closed c o n v e x cone w i t h v e r t e x r T h e n X + Y = cl c o n y ( X U Y).
W i t h C E X N Y, i t is clear t h a t X + Y ~ X U Y, a n d t h e n since X + Y is closed a n d c o n v e x i t follows t h a t X § Y ~ cl c o n y (X U Y). T h e r e v e r s e inclusion s t e m s f r o m t h e f a c t t h a t if x E E a n d e is a r a y e m a n a t i n g f r o m r t h e n x + ~ c cl c o n v ({x} U ~).
N o w s u p p o s e B is a b o u n d e d p o l y h e d r o n a n d C is a p o l y h e d r a l cone w i t h v e r t e x ~b.
L e t B ' = c o n y ( B U {~)) a n d B" = B - b for some b E B. F r o m (2) we see t h a t if K = B + C, t h e n K - b = B" + C = el c o n y (B" U C). T h u s (iv) is e q u i v a l e n t t o (v) a n d t h e p r o o f of 2.12 is c o m p l e t e .
2.15. COROLLARY.
I]
K is a polyhedron (a bounded polyhedron, a polyhedral cone), then so is every a/fine image o / K .Proo]. F o r b o u n d e d p o l y h e d r a a n d p o l y h e d r a l cones, t h i s is e v i d e n t b y 2.13 a n d 2.14.
T h e n use 2.12 (v) for t h e g e n e r a l case.
2.16. COROLLARY. I / K~ and K 2 are polyhedra (bounded polyhedra, polyhedral cones with vertex r then so are the sets K 1 + K 2 and cl conv ( K 1 U K~).
Proo]. F r o m 2.13 i t follows t h a t if B 1 a n d B 2 a r c b o u n d e d p o l y h e d r a , t h e n so a r e B~ + B 2 a n d conv (B~ U B~). F r o m 2.14 i t follows t h a t if C 1 a n d C 2 a r e p o l y h e d r a l cones, t h e n C 1 + C~ is a p o l y h e d r a l cone, as is c o n y (C 1 U C~) u n d e r t h e a d d i t i o n a l a s s u m p t i o n t h a t C 1 a n d C s h a v e t h e s a m e v e r t e x . T h u s if K 1 a n d K 2 a r e p o l y h e d r a l ( s a y K~ = B~ + C~, so t h a t K 1 + K 2 = (B~ + Bz) + (C 1 + C~)), we d e d u c e f r o m 2.12 (v) t h a t K 1 + K~ is p o l y -
86 VICTOR K L E E
hedral. Observe, finally, t h a t when K 1 a n d K2 are polyhedral it is evident from 2.12 (iii) t h a t the set K = cl conv (K 1 U K2) is the closed convex hull of a finite system of points and rays; t h a t K is polyhedral is then a consequence of (1) above in conjunction with 2.11.
The proof of 2.16 is complete.
I n concluding this section, we mention two more notions which play an i m p o r t a n t role in the sequel. A subset K of E is said to be boundedly polyhedral provided its intersection with each bounded polyhedron in E is polyhedral, a n d to be polyhedral at a point p E K provided some neighborhood of p relative to K is polyhedral.
2.17. PROPOSITIOn. A set is boundedly polyhedral i// it is closed, and convex, poly- hedral at all its points.
Proo/. Suppose K is closed, convex, and polyhedral a t all its points, and consider a bounded polyhedron B. E a c h point x E K admits a bounded polyhedral neighborhood/V~
relative to K, and b y compactness of B N K there must be a finite set X ~ K with B N K c IJ N,. L e t Z = cony IJ N,. Then Z is polyhedral, whence so is B (1 Z, and we have B N K
x ~ X x ~ X
Z ~ K. I t follows t h a t B N K = B N Z and K m u s t be boundedly polyhedral.
3. Cones and polyhedra
For a point p E E and a set X c E, cone (p, X) will denote the s e t p + ] 0 , o o [ ( X - p ) , the smallest cone which contains X and has vertex p. (Note t h a t p Econe (p, X ) i f f p E X . ) The present section consists largely of exploiting the connection b e t w e e n t h i s notion and polyhedra. We begin with a collection of elementary but useful facts a b o u t convex cones, supplying some of the machinery to be employed in proving the m a i n theo- rems.
3.1. P R O r O S I T I O ~ . Suppose C is a convex cone in E with vertex r and let L denote the lineality space o/ cl C (L = cl C N - c l C). Then C is linear i// C ~ Z. Suppose now that C is not linear, so C dg L. Then there is a linear/unctional / on E such that / = 0 on L and / > 0 on cl C ,,~ L. With t > 0, let H o = / - 1 0 and Ht = / - 1 t. Then the/ollowing statements are true:
(i) C = (H 0 N C) U ]0, oo[(Ht N C);
(ii) H t N C D ( H 0 N C ) + ( H t N C ) and C ~ ( H 0 N C ) + [ 0 , ~ [ ( H t N C ) , with equality when r E C (and also under certain other conditions);
(iii) i/ p E H t N C, then cone (p, Ht N C)=7eC + p , where ~ is the projection o] E onto H o which is the identity on H o and maps p onto r
(iv) i / S is a subspace supplementary to L in H o and St is a translate o / S to H, then St N C is bounded;
SOME C H A R A C T E R I Z A T I O N S OF C O N V E X F O L Y H E D R A 87 (v) i / L c C, then Ht N C = L + (St fl C) and C = L + [0, ~ [ ( S t N C);
(vi) C is closed i]/ Ht N C is closed and L c C;
(vii) C is polyhedral i/[ Ht N C is boundedly polyhedral and L ~ C.
Proo[. T h e existence of / as described is well-known. Since / > 0 on C ~ L, it is clear t h a t each point of C ~ L has a positive multiple in Ht N C, whence condition (i) holds. I f H 0 N C = A, t h e inclusions of (ii) hold trivially, for always A + X = A. Suppose on t h e other h a n d t h a t x E H0 N C a n d y E Ht N C. T h e n x + y E C (since C + C c C) a n d / (x + y) = 0 + t, so x + y E H t N C. This justifies t h e first inclusion in (ii). F o r t h e second, we see f r o m (i) t h a t C ~ ]0, oo[(Ht N C), whence C D ]0, c~ [{(H 0 N C) + (Ht N C)} = ( H 0 N C) + ]0,c~[.
(Ht N C). A n d , b y (i), C ~ H 0 N C = (H 0 N C) + {0} (Ht N C), so t h e second inclusion of (ii) is established. The assertions a b o u t e q u a l i t y in (ii) are easily verified.
N o w to establish (iii), let K = (Ht N C) - p, so t h a t K c H 0 a n d Ht N C = K + p. T h e n
a n d
cone (p, Ht N C) ~ cone (p, K + p) = p + ]0, ~ [K C = z ( H o N C) U ~ ] 0, ~ [ ( K + p) = ( H o N C) c ] 0 , r162 [ K .
F r o m (ii) we see t h a t K ~ (H o N C) + K, whence K D H o n C (since r E K) a n d ] 0, ~ [ K ~ ] 0,
~ [ ( H o N C) = H o N C. I t follows t h a t z C = ]0, ~ [ K a n d (iii) has been proved.
F o r (iv), observe t h a i if St N C is u n b o u n d e d it m u s t contain a r a y a n d t h e n t h e parallel r a y e m a n a t i n g f r o m r m u s t lie in S N cl C, contradicting t h e fact t h a t H o N cl C = L.
N o w if L c C, t h e n L + C c C a n d hence L + (St N C) c Ht N C. T h e reverse inclusion follows f r o m t h e fact t h a t Ht = L + St, a n d t h e n using (ii) we o b t a i n
C = L + [0, ~ [ ( L + (St N C)} = L + [0, ~ [ ( S t N C), so (v) is proved.
I t is easy to verify t h a t if Ht N C is closed, so is Is, ~ [(Ht N C) for each s > 0. A n d since H 0 N cl C = L, it follows t h a t C is closed when Ht N C is closed a n d L ~ C. This establishes (vi).
N o w suppose Ht N C is b o u n d e d l y p o l y h e d r a l a n d L c C. Since St N C is bounded, it is t h e intersection with Ht N C of a b o u n d e d p o l y h e d r o n in St, a n d t h u s St N C is polyhedral.
W i t h V denoting t h e set of all extreme points of St N C a n d B a basis for L, it is clear t h a t C consists of all non-negative combinations of elements of B U - B U V, for C = L + [0,
~ [ ( S t N C) b y (v). Thus C is p o l y h e d r a l a n d t h e proof of 3.1 is complete.
3.2. P R O r O S I T I O N . Suppose X and Y are convex subsets o / E , p E X n Y, and y c cone (p, X). Then i/ Y is polyhedral, the set X N Y is a neighborhood o / p relative to Y.
8 8 VICTOR K L E E
Proo/. L e t L a n d S~ be as in 3.1, so t h a t C = L + [0, c~[(St N C). W e assume w i t h o u t loss of generality t h a t p = r N o w Y can be expressed as t h e intersection of closed half-
k
spaces Q1 . . . Qk, Qk+l . . . Qm such t h a t ~bE int Q~ iff k + 1 ~<i ~< m. L e t C = f i Qi. T h e n
1
y c C c cone (p, X) a n d C is a p o l y h e d r a l cone with v e r t e x r To prove 3.2 it suffices to prove t h a t X N C is a n e i g h b o r h o o d of r relative to C. Since St N C is a b o u n d e d polyhedron, there is a finite set V such t h a t c o n y V = St N C, a n d t h e n for each v E V there exists rv > 0 such t h a t [r rvv] ~ X . The n u m b e r r = inf (rv : v E V} is positive a n d [0, r] V ~ X, whence [0, r] (St N C) ~ X. Observe also t h a t r is an inner point of T N X for each line T ' t h r o u g h r in L, a n d hence X N L is a n e i g h b o r h o o d of r relative to L. (We are using here t h e well-known fact t h a t 3.2 is valid when Y = L . ) N o w each point u of C has a unique expression in t h e f o r m u = w~ + a~z~ with wu EL, au E [0, c~ [, a n d z~ E St N C. Clearly there is a n e i g h b o r h o o d N of r such t h a t w u E l ( X N L) a n d auE[0, i r] for all u E N N Y; we t h e n
h a v e
N n Y ~ 89 n L) + 89 r](St N C ) ~ c o n y X = X, a n d t h e proof of 3.2 is complete.
3.3. COROLLARY. A convex set K is polyhedral at a point p E K i// cone (p, K) is polyhedral.
Proo/. I f cone (p, K) is polyhedral, it follows f r o m 3.2 t h a t K is a n e i g h b o r h o o d of p relative to cone (p, K). B u t t h e n p has a p o l y h e d r a l n e i g h b o r h o o d N in E such t h a t N N cone (p, K) = P c K, a n d P is a polyhedral n e i g h b o r h o o d of p relative to K.
Conversely, if K is p o l y h e d r a l at p there is a b o u n d e d p o l y h e d r a l n e i g h b o r h o o d J of p relative to K. F r o m c o n v e x i t y of K it is clear t h a t cone (p, K) = cone (p, J), which is obviously polyhedral.
3.4. COROLLARY. The result 3.2 is valid under the assumption that X is polyhedral rather than Y.
Proo]. If X is polyhedral, t h e n cone (p, X) is p o l y h e d r a l b y 3.3, a n d it follows f r o m 3.2 t h a t X is a n e i g h b o r h o o d of p relative to cone (p, X). W i t h Y c cone (p, X), this is sufficient.
3.5. COROLLARY. 1/ p is a point o / a convex set K, then cone (p, K ) is closed i// J N K is polyhedral at p / o r each 2-/lat J through p.
Proo/. Use 3.3 in conjunction with t h e following two facts: a two-dimensional c o n v e x cone is p o l y h e d r a l iff it is closed; a convex set is closed iff all its 2-sections t h r o u g h a given p o i n t are closed.
SOME C H A R A C T E R I Z A T I O N S OF C O N V E X P O L Y H E D R A 89 W e c o n c l u d e t h i s s e c t i o n w i t h t w o basic l e m m a s w h i c h will b e s u b s u m e d b y t h e t h e o r e m s of w 4.
3.6. LEMMA. Suppose 2 <~ ~ <<. n and K is a convex subset o / E n all o/whose ~-projections are closed. Then i / a l l 2-sections o / c l K are boundedly polyhedral, X must be closed.
Proo/. W e m a y a s s u m e t h a t • E K , j < n, a n d l e t L d e n o t e t h e u n i o n of all lines t h r o u g h r w h i c h lie in c] K , M a s u b s p a c e s u p p l e m e n t a r y t o L in E. T h e n ( b y 2.7) cl K = L + M N (el K ) , w h e r e t h e set A = M N el K c o n t a i n s no lines a n d t h u s b y 2.3 A = c o n y (ex A U r e x A), w h e r e ex A is t h e set of a l l e x t r e m e p o i n t s of A a n d r e x A t h e u n i o n of its e x t r e m e r a y s . C l e a r l y A i n h e r i t s f r o m cl K t h e p r o p e r t y t h a t a l l its 2-sections a r e b o u n d e d l y p o l y h e d r a l , a n d we can show t h a t K is closed b y showing t h a t L + ex A c K a n d L + r e x A c K .
W e use t h e following t w o facts: (a) L + r e l i n t A c K ; (b) if J = {p} for p E ex A or J = cl ~ for some e x t r e m e r a y ~ of A , t h e n A is s u p p o r t e d b y a h y p e r p l a n e H in M such t h a t H N A = J . T h e a s s e r t i o n (a) follows f r o m t h e r e a d i l y v e r i f i e d f a c t s t h a t e a c h finite- d i m e n s i o n a l c o n v e x set c o n t a i n s t h e i n t e r i o r of its closure, a n d L + r e l i n t A c r e l i n t cl K . T h e a s s e r t i o n (b) is a n e a s y consequence of 2.17, 3.5, a n d t h e e x i s t e n c e of [ a s d e s c r i b e d in 3.1.
Consider first a n e x t r e m e p o i n t p of A . T h e a s s e r t i o n (b) g u a r a n t e e s t h e e x i s t e n c e of a s u p p o r t i n g h y p e r p l a n e U of K such t h a t U N cl K = p + L. A n d if p + L ~= K , t h e r e m u s t b e a h y p e r p l a n e V in U b o u n d i n g a n o p e n h a l f s p a c e W in U such t h a t W misses (p + L) N K b u t i n c l u d e s a p o i n t w of p + L. L e t X be a (j - 2 ) - d i m e n s i o n a l f l a t in V, x a p o i n t of X , q Erelint A a n d Y t h e j - d i m e n s i o n a l f l a t c o n t a i n i n g X U {q, w}. L e t ~ be a n affine p r o j e c t i o n of V o n t o X . E a c h p o i n t z E E h a s a u n i q u e e x p r e s s i o n in t h e f o r m
z = x + (z 1 - x) 4- z 2 (w - x) + z a (q - x) w i t h z 16 V, z~ E R, za E R , a n d for e a c h z we define
~z = x + (~z l - x ) + z~(w - x ) + z3(q - x ) ,
so t h a t v/is a n affine p r o j e c t i o n of E o n t o Y. B y h y p o t h e s i s , ~ K is closed. N o w if z 6 K ~ U, t h e n za > 0 a n d ~ z # w; if zE U N K , t h e n z2 ~< 0 a n d ~z4= w. T h u s w ~ ? K . B u t ~/]w, q] = ]w, q] c K , so i t follows t h a t w Ecl ~/K -~ 2/K, a n i m p o s s i b i l i t y . W e c o n c l u d e t h a t p + L c K for e a c h p E e x A .
N o w consider a n e x t r e m a l r a y ~ of A - - s a y e = P + ]0, c o [ ( u - p) where u 6 ~ a n d p is t h e e n d p o i n t of ~. B y (b), t h e r e is a s u p p o r t i n g h y p e r p l a n e U of K such t h a t U N e l K = cl ~ + L . Since p E e x A , t h e r e s u l t of t h e l a s t p a r a g r a p h shows t h a t p + L c K , a n d i t fol-
90 V I C T O R K L E E
lows easily t h a t (q + L) N K = a + L for some "initial s e g m e n t " a of ~. W e wish to show t h a t a = ~ . Suppose not, a n d let wE~ ~ cl~. I t can be verified t h a t U m u s t contain a h y p e r p l a n e V bounding a n open halfspace W in U such t h a t w E W b u t W misses (Q + L) N K.
A contradiction is reached as in t h e preceding p a r a g r a p h , a n d t h e proof of 3.6 is complete.
3.7. LEMMA. Suppose 2 < ?, ~ n, and C is a convex cone in E n all o / w h o s e ?,-pro?,ections are closed. T h e n all (n - j + 1)-sections o / C are polyhedral.
Proo/. Application of 3.1 (vii) shows t h a t all r-sections of a c o n v e x cone C are p o l y h e d r a l iff M N C is p o l y h e d r a l for e v e r y (r + 1)-flat M t h r o u g h t h e v e r t e x of C. This equivalence will be used in t h e p r e s e n t proof, a n d we refer to it as (**).
N o w consider a fixed j ~> 2, a n d let hrj d e n o t e t h e set of all integers n / > ?, such t h a t each c o n v e x cone in E n which has all its ?.-projections closed m u s t also h a v e allits (n - ?, + 1)- sections polyhedral. Clearly ?, ENj. N o w suppose k - 1 ENs, a n d consider a c o n v e x cone C w i t h v e r t e x r in E k, all j-projections of C a s s u m e d t o b e closed. W e wish t o show t h a t all (k - ?, + 1)-sections of C are polyhedral, a n d observe t h a t it suffices to do this for cl C, for t h e n it follows b y 3.6 t h a t C is closed.
L e t L, H0, a n d Ht be as in 3.1. Consider a n a r b i t r a r y p o i n t p E H t N cl C, a n d let zr be as in 3.1 (iii). T h e n 7r e 1C is a c o n v e x cone in t h e (/c - 1)-dimensional space H 0 a n d all its j-projections are closed (being j-projections of C), so f r o m t h e i n d u c t i v e h y p o t h e s i s it follows t h a t all (k -?.)-sections of ~relC are polyhedral. N o w 3.1 (iii) shows t h a t ~rclC is m e r e l y a t r a n s l a t e of cone (p, Ht N cl C), so all (k - ])-sections of t h e l a t t e r are polyhedral.
Using (**) a b o v e we see t h a t cone (p, J N cl C) is p o l y h e d r a l for each (k - ?" + 1)-flat J t h r o u g h p in Hr. A n application of 3.3 shows t h a t for each (k - ?, + 1)-flat Gt in Ht, t h e set Gt N cl C is p o l y h e d r a l a t each of its points, a n d hence b y 2.17 m u s t b e b o u n d e d l y polyhedral. F r o m 3.1 (vii) we conclude t h a t t h e cone cl [0, ~o [(Gt N C) is p o l y h e d r a l a n d hence t h a t Gt N cl C is polyhedral. Since Gt is a n a r b i t r a r y (k - ?, + 1)-flat in Ht, it fol- lows f r o m 3.1 (vii) a n d (**) t h a t all (k - ?, + 1)-sections of el C are polyhedral. T h u s C is closed b y 3.6 a n d all (k - ])-sections of C are polyhedral.
I n t h e preceding two p a r a g r a p h s we showed t h a t if k - 1 ENj, t h e n k E N ] . I t follows b y m a t h e m a t i c a l induction t h a t N j includes all integers ~> ?,, a n d 3.7 has been proved.
4. Projections and sections W e s t a r t w i t h t h e f u n d a m e n t a l
4.1. THEOREM.
Suppose
K is a convex subset o/ E n, p E K , and 2 <~?, <~n. T h e n K is polyhedral at p i / / xrK is polyhedral at p whenever ~r is an a/line projection o / K onto a ?,-/lat through p.SOME C H A R A C T E R I Z A T I O N S O F CONVEX P O L Y H E D R A 9 1
Proo/. F o r each re as described, it is evident t h a t cone (p, reK) = re cone (p, K). T h e n if K is p o l y h e d r a l at p, cone (p, K) is p o l y h e d r a l b y 3.3, whence of course re cone (p, K) is p o l y h e d r a l a n d a second application of 3.3 shows t h a t reK is p o l y h e d r a l at p. On t h e other hand, if reK is p o l y h e d r a l at p for each re as described, t h e n all j-projections of cone (p, K) are polyhedral, whence all 2-projections are polyhedral. I t follows b y 3.7 t h a t all (n - 1)- sections of cone (p, K) are polyhedral, whence cone (p, K) is itself p o l y h e d r a l b y 3.1 (vii).
W e conclude f r o m 3.3 t h a t K is p o l y h e d r a l at p, a n d t h e proof of 4.1 is complete.
4.2. COROLLARY. With 2 <~ j <~ n, a convex subset o / E n is polyhedral at all its points i// all its ].projections have this property.
4.3. COROLLARY. With 2 <~ j <--. n, a closed convex subset o/ E n is boundedly polyhedral i// all its j-projections are polyhedral at each point.
4.4. COROLLARY. W i t h 2 <~ j <~ n, i/ all j-projections o/ a convex subset o/ E ~ are boundedly polyhedral, the set itsel/ must be boundedly polyhedral. Conversely, each closed j- projection o/ a boundedly polyhedral set in E n is boundedly polyhedral (but o / c o u r s e there m a y be j-projections which are not closed).
4.5. COROLLARY. With 2 <~ j <. n, a bounded convex subset o/ E ~ is polyhedral i// all its j-projections are polyhedral.
I n w 5 we c o n s t r u c t a n o n p o l y h e d r a l convex set in E 3 all of whose 2-projections are polyhedral. (The set is necessarily u n b o u n d e d a n d b o u n d e d l y polyhedral.) Thus for j = 2, t h e restriction to b o u n d e d sets is essential in 4.5, b u t we shall see below t h a t for j >~ 3 t h e restriction can be removed.
Since we k n o w a l r e a d y t h a t all j-projections a n d j-sections of a p o l y h e d r o n are poly- hedral, a n d all j-sections of a b o u n d e d l y p o l y h e d r a l set are b o u n d e d l y polyhedral, t h e remaining results of this section are m o s t simply s t a t e d n o t in terms of a range of values for j, b u t instead in terms of t h e " b e s t " value for j.
W e need t h e following
4.6. REMARK. Suppose K is a closed convex set in E with r L is a subspace o / E , and re is a linear projection o/ E ' whose kernel is L ~ Then (L O K ) ~ = L ~ + cl reK ~ = re-1 (cl reK~
Proo[. Since L ~ is a subspace of E ' a n d reK ~ lies in t h e s u p p l e m e n t a r y subspace reE', it is easy to check t h a t
(L N K) ~ = cl c o n y (L ~ U K ~ = el (L ~ + K ~
= el (L ~ + reK ~ = L ~ + el reK ~ = re-1 (el reK~
9 2 VICTOR KLEE
4.7. THEOREM. I / K is convex and p E i n t K , then K is polyhedral [reap. boundedly polyhedral] ill all its 2-sections through p are polyhedral [reap. boundedly polyhedral].
Proo/. Obviously K is closed and we m a y assume t h a t p = r whence K = K ~176 and K ~ is bounded. Now consider a linear projection g of E' onto one of its 2-subspaees a n d let M be the kernel of ~, L = M 0. Then L ~ = M and we see from 4.6 t h a t (L N K) ~ = M + cl H K ~ But K ~ is compact, so HK ~ is closed, and thus if L n K is polyhedral it follows t h a t M + H K ~ is polyhedral, whence H K ~ is itself polyhedral. Thus if all of K ' s 2-sections through r Eint K are polyhedral, all 2-projections of K ~ are polyhedral. From 4.5 it follows t h a t K ~ is a polyhedron, and then b y 2.11 t h a t K is one. Thus 4.7 has been proved for polyhedra, and it remains only to consider the case of boundedly polyhedral sets.
Suppose all 2-sections of K through p eint K are boundedly polyhedral, and consider an arbitrary bounded polyhedron B. There is a bounded polyhedron Q ~ B such t h a t p e i n t Q; of course all 2-sections of Q N K through p are polyhedral and hence Q N K is polyhedral b y the result just established. B u t then B N K must be polyhedral, so K is boundedly polyhedral and the proof of 4.7 is complete.
4.8. COROLLARY. I] K is convex and
geE,
then K is polyhedral [reap. boundedly polyhedral] i// all its sections by 3-/fats through q are polyhedral [resp. boundedly polyhedral].4.9. THEOREM. A convex set K c E n {with n >~ 3) has polyhedral closure i]] all its 3- proiections have polyhedral closure.
Proo[. Assume r e K. Then if all 3-projections of K have polyhedral closure, it follows b y 4.6 that all 3-sections of K ~ through r are polyhedral, whence K ~ is polyhedral b y 4.8 and the set cl K = K ~176 is polyhedral b y 2.11.
4.10. COROLLARY. A convex set K c E n {with n >~ 3) is polyhedral ill all its 3-projec- tions are polyhedral.
Proo/. Use 4.9 and 3.6.
4.11. TttV, OR]~M. With 2 ~ < ] ~ < n - l, a convex cone in E n is polyhedral i]] all its l- projections are closed.
Proo[. Let C be the cone in question, q a vertex of C. If all ]-projections of C are closed, then all ( n - ] + 1)-sections of U are polyhedral b y 3.7, whence all 2-sections of C are polyhedral. Each 3-section of U through g must then be polyhedral by 3.1 (vii), and from 4.8 we conclude t h a t C is polyhedral.
We wish to deduce from 4.11 a result on the extension of positive linear functionals. The connection between projections and extensions m a y be stated as follows:
S O M E C H A R A C T E R I Z A T I O N S O F C O N V E X P O L Y H E D R A 93 4.12. REMARK. Suppose C is a closed convex cone in E with vertex 4, L is a subspace o / E , and ~ is a linear projection o / E ' whose kernel is L ~ T h e n ~ C o is closed i// every linear /unctional on L which is >~ 0 on L N C can be extended to a linear/unctional on E which is >~ 0 on C.
Proo/. For each [ E E ' , let ~ / b e the restriction of [ to L, so t h a t ~ is a linear map of E ' onto L'. The extendability condition above is equivalent to the condition t h a t ~C ~ =
(L N C) ~ or, since L ~ is the kernel of ~, t h a t L ~ + C O = L ~ + (L N C) ~ B u t L ~ + C O = L ~ § ~ C ~ and (L N C) ~ = L ~ + cl z C ~ b y 4.6. Thus the desired conclusion follows.
The following consequence of 4.11 and 4.12 was proved b y Mirkil [7] for ] = 2:
4.13. T~IEOREM. Suppose C is a closed convex cone with vertex r in E n, and let us say tha~ C has the property Pk (/or 0 <- k <~ n) i// every linear/unctional on a k-subspace L o]
E n which is >1 0 on L N C can be extended to a linear/unctional on E n which is >1 0 on C.
T h e n C must have the properties Po, 1)1, and Pn; b u t / o r 2 <~ ~ <~ n - l, C has property P j i//
C is polyhedral.
5. Some examples and further results
This section contains a rather discursive treatment of material which was suggested b y the results and methods of earlier sections, but was not essential in dealing with the principal theorems in w 4. We discuss primarily the sets which are boundedly polyhedral and those which are polyhedral away from • (a notion defined below), characterizing these in various ways and describing their polar sets. We construct in E 3 a nonpolyhedral set K ~ such that both K and K ~ are boundedly polyhedral, and also a nonpolyhedral convex set all of whose 2-projections are polyhedral.
I n analogy with earlier definitions, a convex set K is said to be polyhedral at ~ iff each polyhedron (or, equivalently, each closed halfspace) in E has a translate whose intersection with K is polyhedral, and K is said to be polyhedral a w a y / r o m the point p Ecl K iff K has polyhedral intersection with each polyhedron in E ~ (p}. The following two results should be compared with 2.17.
5.1. PROPOSITIOn. A convex set K is polyhedral i// it is closed and is polyhedral at each point o / K U (c~ ~.
Proo/. Let F be a basis for the conjugate space E'. Then if K is polyhedral at ~ , there exists t > 0 such t h a t each of the sets P~ = / - 1 ] _ ~ , _ t] N K and Qr = / - 1 It, ~ [ N K is polyhedral. And if K is closed and is polyhedral at each point of K, then K is boundedly
94 VICTOR KLEE
p o l y h e d r a l b y 2.17 a n d hence its intersection with t h e set f l / - 1 [ _ t, t] is a p o l y h e d r o n B.
fEF
B u t of course K = cl cony (B 0 (UP1) U (UQI)), a n d it follows b y 2.16 t h a t K is polyhedral.
fcF IEF
5.2. COROLLARY. I / K is convex and p e e l K, then K is polyhedral a w a y / r o m p i//
el g c K 0 ( p ) and K is polyhedral at each point o / ( K 0 ( ~ )) ,,~ ( p ) .
Proo/. T h e " o n l y if" p a r t follows at once f r o m t h e fact t h a t each point of E ~ ( p ) has a p o l y h e d r a l n e i g h b o r h o o d in E which misses p. F o r t h e "if" part, consider a poly- h e d r o n J in E N (p). T h e hypotheses i m p l y t h a t J N K is closed a n d is p o l y h e d r a l at each p o i n t of ( J N K) U ( o o ) , whence application of 5.1 completes t h e proof.
W e see f r o m 2.16 t h a t t h e convex hull of t h e union of two p o l y h e d r a is p o l y h e d r a l if it is closed. I n addition to t h e obvious result for b o u n d e d polyhedra, we note
5.3. P R O P O S I T I O N . I] X and Y are polyhedral cones each o / w h i c h contains a vertex o / t h e other, then conv ( X 0 Y) is polyhedral.
Proo/. L e t p be a v e r t e x of X in Y, q a v e r t e x of Y in X, a n d let t h e origin in E be so chosen t h a t q = - p. T h e n there are finite subsets U a n d V of E such t h a t - p E U, p E V, X - p consists of all non-negative combinations of U, a n d Y - q consists of all n o n - n e g a t i v e combinations of V. B u t t h e n it is easy to verify t h a t c o n y (X U Y) consists of all non- negative combinations of U (J V.
T h e following result has some useful corollaries.
5.4. P ~ O P O S I T I O N . I / X and z are points o/ a convex set K and y e ] x , z [ , then cone (y, K) = c o n v [cone (x, K) U cone (z, K)].
Proo/. W e m a y assume t h a t x =~ z a n d let J = e o n v [cone (x, K) U cone (z, K)]. Consider a point u e cone (y, K), n o t collinear with [x, z]. There exists v e ] u, y [ N K. I t is easy to verify t h a t u lies on a segment joining t h e r a y from x t h r o u g h v t o t h e r a y f r o m z t h r o u g h v, a n d we conclude t h a t cone (y, K) c J .
N o w consider a p o i n t w e ] x , y[, a n d observe t h a t [x, v ] c K a n d In, w[ intersects [x, v], so u e cone (w, K). T h u s cone (y, K) c cone (w, K). A similar a r g u m e n t establishes t h e reverse inclusion, a n d we conclude t h a t cone (s, K) is c o n s t a n t for s e ] x , z[. T h u s to p r o v e t h a t J c cone (y, K) it suffices to show t h a t if p e cone (x, K), q e cone (z, K), a n d c = r p § (1 - r ) q for r e ] 0 , 1[, t h e n e e cone (s, K) for some s e ]x, z[. U n d e r these condi- tions there are positive n u m b e r s ~ a n d fl such t h a t x + [0, ~] (p - x) ~ K a n d z + [0, fl] (q - z) ~ K. W i t h m = m i n (~, fl), p' = x + m ( p - x), q' = z + m ( q - z), a n d s = r x + (1 - r)y, it can be verified t h a t c = s + m -1 [rp' + (1 - r ) q ' - s ] , whence c e cone (s, K) a n d t h e proof of 5.4 is complete.
S O M E C H A R A C T E R I Z A T I O N S O F C O N V E X P O L Y H E D R A 95 5.5. COROLLARY.
If
K is convex and Y is the set o/all points o / K at which K is locally polyhedral, then Y is convex.Proo/. Use 3.3, 5.4, a n d 5.3.
5.6. COROLLARY. I / a nonpolyhedral closed convex set K is polyhedral away /tom p E K , then p is an extreme point of K .
Proo/. Use 5.2 a n d 5.5.
5.7. COROLLARY. I t K is a closed convex set and K = conv S, then K is boundedly polyhedral i]/ K is polyhedral at each point o i S.
Proo]. Use 2.17 a n d 5.5.
I n connection with 5.7, we n o t e
5.8. P R O P O S I T I O N . A closed convex set K is boundedly polyhedral i H cone (p, K) is closed/or each p e K .
Proo/. If cone (p, K) is closed for each p e K , it follows f r o m 3.5 a n d 5.7 t h a t each 2-section of K is b o u n d e d l y polyhedral, whence K m u s t be b o u n d e d l y p o l y h e d r a l b y 4.7.
Comparing 5.8 with 5.7, it is n a t u r a l to ask w h e t h e r a closed convex set K m u s t be b o u n d e d l y p o l y h e d r a l if cone (p, K) is closed for each p o i n t p of a set S such t h a t conv S = K. T h e answer is negative, as t h e following example shows. I n E z, let J be a non- p o l y h e d r a l two-dimensional c o m p a c t c o n v e x set which is p o l y h e d r a l a w a y f r o m $ a n d let S be a segment which has r as an inner point a n d is n o t c o p l a n a r with J . L e t K -- conv (S t) J). T h e n cone (p, K) is closed except when p is an inner p o i n t of S.
T h e following is a n analogue of 5.5.
5.9. P R O P O S I T I O N . Suppose K is a closed convex subset o] E and 14" is the set o / a l l ] e E' such that ]-1 [t, c~ [ N K is polyhedral/or some t < c~. Then F is a convex cone with vertex r
Proo/. Clearly [0, c~ I F c F , a n d it remains to prove t h a t F + F c F . Consider [, g e F , with h = i + ff a n d r, s e R such t h a t t h e sets P = / - 1 [r, c~ [ N K a n d Q = / - 1 [s, oo [ n K are b o t h polyhedral. T h e n h-l[r + s, ~ [ c / - l [ r , c~[ O g-l[s, oo[, a n d it follows t h a t
h -1 [r + s, co [ n K = h -1 [r + s, oo [ n cl c o n y (P U Q).
B u t cl c o n y ( P t3 Q) is p o l y h e d r a l b y 2.16 a n d t h e desired conclusion follows.
W e wish n e x t t o describe t h e polars of b o u n d e d l y p o l y h e d r a l sets. I n doing this we e m p l o y t h e following proposition, which goes a bit b e y o n d our i m m e d i a t e use for it.
5.10. PROPOSlTIOI~. Suppose K is a closed convex set in E, p E K , K contains no line, C = (x : [0, oa Ix ~ K - p}, and F is the set o / a l l linear/unctionals / on E such that / > 0
96 WCTOR KLE~
on C ,,, {~}. T h e n F is a nonempty open convex cone in E ' with vertex ~'. A closed hal/space in E has the form f--if --OO, r] for some f E E and r E R i f / e a c h o / i t s translates has bounded intersection with K. I f C 4 K - p there e x i s t s / E F such that/-1] _ ~ , /p] N K has nonempty interior relative to the smallest flat containing K .
Proof. Clearly C o is a closed c o n v e x cone with vertex ~', a n d since C contains no line C o is n o t contained in a n y hypel~plane in E. B u t t h e n C o m u s t have interior points, a n d it is n o t h a r d to verify t h a t F = - int C ~ p r o v i n g t h e first assertion of 5.10. N o w if a half- space in E has t h e f o r m / - 1 ] _ c~, r], t h e n each of its translates has the f o r m ]-1] _ oo, s], a n d for f E F it is evident t h a t t h e convex set ]-1] _ ~ , s] N K contains no r a y a n d hence is bounded. On t h e other hand, if f E E ' ~ F t h e n / - 1 ] _ o ~ , / p ] N K is easily seen to be un- bounded. I t remains to prove the last assertion of 5.10.
I f C 4 K - p there m u s t be a point y E r e l i n t K ~ - (p} such t h a t K does n o t contain the entire r a y from p t h r o u g h y. We m a y assume w i t h o u t loss of generality t h a t y = ~, whence C = {x: [0, ~ [ x c K}. T h e choice of y assures t h a t p ~ - C , so either p E C or R p N C = {r B y a separation t h e o r e m for convex cones there exists ] E F such t h a t p E C or ]p = 0 , a n d t h e n from t h e fact t h a t y E relint K it follows readily t h a t the interior o f / - 1 ] _ o o , / p ] N K relative to fl K is n o n e m p t y .
5.11. PI~OPOSlTION. For a closed convex set K c E with r E K , the/ollowing assertions are equivalent:
(i) K is boundedly polyhedral;
(if) if / is a linear functional on E such t h a t / x > 0 whenever [0, oo[x ~ K but ] - oo, 0]x~= K then/-1] _ oo, s] N K is polyhedral/or each sE R;
(iii) there exists a linear/unctio~ml / on E such that ]-1] _ or s]N K is polyhedral for each s E R;
(iv) el c o n y ( K ~ U N ) is polyhedral/or each polyhedral neighbortwod iV o / r in E' ; (v) cl c o n y (K ~ U [r - g ] ) is polyhedral/or each g e relint K~
(vi) there exists f e E ' such that cl c o n y (K ~ U [r t/f) is polyhedral/or each t > O.
Proof. I t is evident t h a t conditions (i) a n d (iv) are dual u n d e r polarity, a n d hence equivalent b y 2.8 a n d 2.11, as are (iii) a n d (vi). N o w let L a n d M be as in 2.7, so t h a t K = L + (M N K) a n d M N K contains no line, a n d let C = {x: [0, oo [x c M N K}. I t can be veri- fied t h a t / is as described in condition (if) a b o v e iff / > 0 on C ~- {r so it follows f r o m 5.10 t h a t (i) implies (if). B y f u r t h e r use of 5.10 we see t h a t (if) implies (iii) a n d t h a t (if) a n d (v) are dual u n d e r polarity. Since obviously (iii) implies (i), the proof cf 5.11 is complete.
The condition (vi) of 5.11 is m u c h less restrictive t h a n m i g h t at first be imagined.
Indeed, f r o m 4.6 a n d our later result 6.2 on t h e projections of b o u n d e d l y p o l y h e d r a l sets,
S O M E C H A R A C T E R I Z A T I O N S O F C O N V E X P O L Y H E D R A 9 7
it follows t h a t ff k < n a n d X is a n a r b i t r a r y k-dimensional closed convex subset of E n with C E X , t h e n ' t h e r e exists in E n a closed c o n v e x set K 3 r such t h a t K ~ is b o u n d e d l y p o l y h e d r a l a n d X is a k-section of K. T h e following example is also of interest in this con- nection.
5.12. EXAMPLE. There is i n E a a n o n p o l y h e d r a l set K 9 ~ such that both K and K ~ are boundedly polyhedral.
Proo/. We shall c o n s t r u c t a n o n p o l y h e d r a l set W ~ ( - 1, 0, 0) such t h a t W is b o u n d e d l y polyhedral a n d cl c o n y ( { ( - 1 , 0, - t ) } tJ W) is p o l y h e d r a l for each t > 0 . F r o m 5.11 it follows t h a t t h e set K = W + (1, 0, 0) has the s t a t e d properties.
L e t / , g a n d h be sectionally linear positive convex functions on [ - 1, 0[, ]0, 1], a n d ]0, 1] respectively such t h a t / ( - 1) = 0 = g 1, l i m / s = oo = lim g t, a n d l im h t = 0 = l i m h ~ t,
s-.~O t--~O t--~O t--~O
where h + is t h e r i g h t - h a n d derivative of h. ( T h u s / , g, a n d h are all n o n p o l y g o n a l b u t are p o l y g o n a l a w a y f r o m 0.) L e t S' = {(s, ]s, O) : s E [ - I, 0[} a n d T ' = {(t, gt, ht) : t~]0, 1]}.
L e t Q be t h e plane q u a d r a n t consisting of all x = (x 1, x z, x z) E/~ 3 for which x I = 0 a n d x 2~>0~<x z, a n d let S = S ' + Q , T = T ' + Q . I t can be verified t h a t b o t h S a n d T are b o u n d e d l y polyhedral. F o r example, if m < ~ , t h e n the intersection of T with t h e half- space {x : x ~ ~< m) is the closed convex hull of the union of t h e (finitely m a n y ) sets of t h e f o r m {x : x 2 ~< m} fl ((t, gt, hv) + Q ) for t such t h a t gt <. m a n d t = 1, gt = m , or g or h has a corner at t.
N o w let W = c o n v ( S U T ) = S U c o n v ( S ' U T ) . T h e n W N ( x : x 3 = 0 } is t h e non- p o l y h e d r a l set of all points (s, a, 0) for which s E [ - 1, 0[ a n d a >~/s, so it follows t h a t W is nonpolyhedral. To show t h a t W is closed, we consider an a r b i t r a r y point wEcl W. There exist sequences 2~ in [0, 1], s~ in [ - 1, 0[, t~ in ]0, 1], a n d p~ a n d q~ in Q such t h a t (1 - 2~) ((s~,/s~, 0) + p~) + 2~ ((t~, gt~, ht~) + q~)--->w, ~t~-->2 E [0, 1], s~-->s E [0, 1], a n d t~-->t E [0, 1].
L e t z~ = (1 - 21)pt + 2~qf EQ. N o w if s < 0 < t, t h e n ]s~-->/s, 9t~-->gt, ht~--->ht, a n d it follows t h a t z~--->w - (1 - 2 ) ( s , / s , 0) - 2(t, gt, ht). D e n o t i n g this limit b y z, we have z E Q a n d
w = (1 - 2)((s, ls, o) + z) + 2 ((,t, gt, ht) + z) e W .
I f t = 0, t h e n gt~--~ oo; since a l w a y s / s ~ > 0, and/s~---> ~o if s = O, we conclude t h a t 2 = 0 a n d s < 0. N o w with q~ = (0, gtf, ht~) EQ, t h e distance f r o m q~ to t h e point (t~, gti, hti) tends to 0 as i--> oo, a n d it is easily verified t h a t z~ + 2~q~--->w - ( s , / s , 0), whence w e ( s , / s , O) + Q c S ~ W. A similar a r g u m e n t applies when s = 0, a n d we conclude t h a t W is closed.
N o w to complete t h e proof we m u s t show W is b o u n d e d l y p o l y h e d r a l a n d cl conv ({( - 1, 0, - t)} U W) is polyhedral for each t > 0. I n view of 2.17, 5.5, a n d the fact t h a t W is closed, it suffices for the former to show t h a t W is polyhedral at each point of S U T,
7 - 593804. A c t a m a t h e m a t / c a . 102. I r n p r i m 6 le 28 s e p t o m b r e 1959
98 VICTOR K L E E
or e q u i v a l e n t l y t h a t cone (p, W) is p o l y h e d r a l for each p E S U T. Consider a n a r b i t r a r y p E S . I t can be verified t h a t sup {x ~ : xE T ~ cone (p, S)} < c~, whence follows t h e existence of a b o u n d e d p o l y h e d r o n B such t h a t T N cone (p, S) c B + [0, ~ [z c T, with z = (0, 0, 1). L e t U = cone (p, S) a n d V = {p} U cone (p, B). W e h a v e
cone (p, W) = cone (p, c o n y (S U T)) = c o n y [cone (p, S) U cone (p, T)]
= c o n y [U U cone (p, B + [0, ~o [z)] = c o n y [U U (V + [0, o~ [z)].
I t can be verified t h a t for a n y c o n v e x sets X a n d Y, a n d convex cone Z w i t h v e r t e x r conv [(X + Z) U Y] U ( Y + Z) = c o n y [X U ( Y + Z)] U (X + Z).
Applying this with X = U, Y = V, a n d Z = [0, o~ [z, we conclude t h a t
cone (p, W) U (U + [0, c~ [z) = c o n v [ ( U + [0, ~ [z) U V] U ( V + [0, oo [z).
Since evidently U + [0, co [z = U ~ cone (p, W), this reduces t o
cone (p, W) = c o n y (U U V) U (V + [0, oo [z).
N o w V a n d V + [0, oo [z are p o l y h e d r a l b y 2.13, 2.14, a n d 2.16, a n d U is p o l y h e d r a l because p E S a n d S is b o u n d e d l y polyhedral. B u t t h e n c o n y (U U l z) is polyhedral, for U a n d V are p o l y h e d r a l cones with c o m m o n v e r t e x p. T h u s cone (p, W) is t h e u n i o n of t w o poly- hedra and, being convex, m u s t be polyhedral. I t follows t h a t W is p o l y h e d r a l a t each point of S. T h e same a r g u m e n t w i t h S a n d T i n t e r c h a n g e d shows t h a t W is p o l y h e d r a l a t each point of T, a n d we conclude t h a t W is b o u n d e d l y polyhedral.
N o w consider a n a r b i t r a r y t > 0 a n d let ut = ( - 1, 0, - t). I t can be verified t h a t if r is a sufficiently small positive n u m b e r (depending on t) a n d J r = cl c o n y ({ut} U {x : x E T, x 1 = r}), t h e n inf {Ix1] : x E (S U T) ~ Jr} > 0, whence there are p o l y h e d r a M a n d N such t h a t S ~ J r ~ M a n d T ~ J r c N. W e t h e n h a v e
cl c o n y ({ut) U W) = cl c o n y ({u~} U S U T) = cl conv ({ut} U Jr U A U B), which is of course p o l y h e d r a l a n d the proof of 5.12 is complete.
5.13. P R O P O S I T I O N . For a closed convex set K c E with t E K , the assertions (ii)-(iv) below are equivalent, are implied by (i), and i m p l y (i) when K is bounded.
(i) K is polyhedral a w a y / r o m r
(ii) cl c o n y (K U [r - x]) is polyhedral/or each x E g ~
{r
(iii) /or each y E E ~ {r sup y K ~ = oo or y-l[t, ~o [ fl K o is polyhedral/or all tE R ; (iv) i / A and B are "opposite" closed hal/spaces in E having the same bounding hyper- plane and K ~ lies in some translate o / A , then B fl K ~ is polyhedral.
