Removable Sets of Support Points of Convex Sets in Banach Spaces

Abstract

ABSTRACT. A corollary of the Bishop-Phelps theorem is that a closed convex subset C of a Banach space can always be represented as the intersection of its supporting closed half-spaces. In this paper an investigation is made of those subsets S of C such that C is the intersection of those closed half-spaces which support it at points of C\S.This will be true for sets S which are "small" relative to C, where smallness can be measured in terms of dimension, density character, or tr-compactness.Suppose that C is a nonempty closed convex subset of a Banach space E. A point x G C is called a support point of C if there exists a nonzero functional / G E* which attains its supremum on C at x. Any such functional is said to be a support functional of C and the set of all support points is denoted by supp C. It is known [1] that the support points of C are always dense in bdry C, the boundary of C, and that the support functional of C are norm dense among those which are bounded above on C. A corollary of the methods used for these results is the fact that C is always the intersection of all those closed half-spaces which are defined by support functionals [1, Corollary 2]. This result is trivial, of course, if C has nonempty interior, since every boundary point of C is a support point. In this case, in fact, it is easily seen that C can be represented as the intersection of those halfspaces which support it at the points of D for any dense subset D of bdry C (see part (iv) of Theorem 1, below). This fact has played a key role in characterizing those generators of Co-semigroups of operators which leave invariant a given closed convex set with interior [2,3]. In considering the extension of his work [2] to more general convex sets, K. N. Boyadzhiev raised the question (in a letter to the author) of whether one could express C as the intersection of those closed half-spaces which support C at some proper subset of supp C. The purpose of this note is to give some answers to this question.A little thought shows that one has to use some care in deleting subsets of supp C. For instance, if C is a line segment, then a point x of F\C on the line determined by C can be separated from C only by those support functionals which attain their maximum on C at the endpoint nearest x; that is, one cannot remove that endpoint from supp C and still separate x from C by a support functional. (If the dimension Of E is at least two, then every point of C is a support point, so supp C minus a single point is still dense in bdry C.) If C is infinite dimensional, then one can remove a finite subset S of supp C; in fact, as we show below, one can remove a finite dimensional subset and still obtain C as the intersection of half-spaces which