### A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

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 approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space $Con{v}_{}\left(X\right)$ of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex...