Removable sets of support points of convex sets in Banach spaces

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 a-compactness. Suppose that C is a nonempty closed convex subset of a Banach space E. A point x E C is called a support point of C if there exists a nonzero functional f E 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 functionals 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 E\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 poinit 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 Received by the editors October 7, 1985 and, in revised form, January 20, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20, 47D05.