Some remarks of drop property


Let C be a proper closed convex set. C is said to have the drop property if for any nonempty closed set A disjoint with C, there is a ∈ A such that co(a, C) ⋂ A = (a). We show that if X contains a noncompact set with the drop property, then X is reflexive. Moreover, we prove that if C is a noncompact closed convex subset of a reflexive Banach space, then C has the drop property if and only if C satisfies the following conditions: (i) the interior of C is nonempty; (ii) C does not have any asymptote, and the boundary of C does not contain any ray; and (iii) every support point x of C is a point of continuity. © 1992 American Mathematical Society.

Publication Title

Proceedings of the American Mathematical Society