Attainable separation property and asymptotic hyperplane for a closed convex set in a normed space

Abstract

Given a closed convex set A in a normed space X, motivated by the classical separation theorem and Gale and Klee's strict separation property, this paper introduces attainable separation property and attainable strict separation property of A. It is proved that if A has attainable separation property then S(A,x⁎)={a∈A|〈x⁎,a〉=supx∈A⁡〈x⁎,x〉} is nonempty and bounded for all x⁎∈X⁎∖{0} with supx∈A⁡〈x⁎,x〉<∞, which reduces to the James theorem if A is bounded. A geometrical notion of an asymptotic hyperplane for A is adopted to study attainable separation properties. It is proved that A has no asymptotic hyperplane if and only if it is continuous in Gale and Klee's sense and that, under the reflexivity assumption on X, A has no asymptotic hyperplane if and only if A−B is closed for every closed convex set B in X with int(B)≠∅. Moreover, it is proved that if X is not reflexive then such difference A−B is not necessarily closed. Using the techniques of variational analysis and in terms of asymptotic hyperplanes, several characterizations and sufficient conditions are established for A to have attainable separation property and attainable strict separation property. In the case of finite dimensional spaces, some sharper results are established.