Locally nonconical convexity

Abstract

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] 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 (Corollary 34). However, there are no infinite-dimensional compact LNC sets (Theorem 23). 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 R^n, and let T be a linear map from R^n into R^m. We show (Theorem 1) 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 (Theorem 3, Corollary 4, and Theorem 5). If Q is strictly convex and T is not 1-1, we can construct continuous sections which take values in the boundary of Q (Theorem 6). 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 (Theorems 16-20). We also prove that T(Q) is LNC (Theorem 25).