Some remarks of drop property

Abstract:

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 $a \in A$ such that ${\text {co(}}a,C{\text {)}} \cap {\text {A = }}\left \{ a \right \}$. 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.