A 'hidden' characterization of approximatively polyhedral convex sets in Banach spaces

摘要

A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ? X at Hausdorff distance < ε from C. We characterize ap- proximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component H of the space ConvH(X) of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if H contains a polyhedral convex set.