Journal of Convex Analysis 08 (2001), No. 1, 087--108
Copyright Heldermann Verlag 2001
Locally Nonconical Convexity
C. A. Akemann
Dept. of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.
G. C. Shell
Department of Defense, 9800 Savage Road, Fort Meade, MD 20755-6000, U.S.A.
Dept. of Mathematics, Washington University, St. Louis, MO 63130, U.S.A.
There is a hierarchy of structure conditions for convex sets. We study a recently defined condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as are half-spaces and finite intersections of sets of either of these types, but many more sets are LNC. For instance, every zonoid (the range of a nonatomic vector-valued measure) is LNC. However, there are no infinite-dimensional compact LNC sets.
The LNC concept originated in a search for continuous sections, and the present paper shows how it leads naturally (and constructively) to continuous sections in a variety of situations. Let Q be a compact, convex set in Rn, and let T be a linear map from Rn into Rm. We show that Q is LNC if and only if the restriction of any such T to Q is an open map of Q onto T(Q). This implies that if Q is LNC, then any such T has continuous sections (i.e. there are continuous right inverses of T) that map from T(Q) to Q, and in fact it is possible to define continuous sections constructively in various natural ways. If Q is strictly convex and T is not 1-1, we can construct continuous sections which take values in the boundary of Q.
When we give up compactness it is natural to consider a closed, convex, LNC subset Q of a Hilbert space X which may be infinite-dimensional. In this case we must assume that T is left Fredholm, i.e. a bounded linear map with closed range and finite-dimensional kernel. We can then prove results analogous to those mentioned in the last paragraph. We also prove that T(Q) is LNC.
Keywords: Convex set, locally nonconical convexity, continuous section, continuous selection, strictly convex set, zonoid.
MSC: 52A20, 46A55; 52A07, 46C05
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