A “hidden” characterization of approximatively polyhedral convex sets in Banach spaces
Volume 210 / 2012
Studia Mathematica 210 (2012), 137-157
MSC: Primary 52A07, 52A27; Secondary 46A55; 46N10, 52B05, 52A37.
DOI: 10.4064/sm210-2-3
Abstract
A closed convex subset $C$ of a Banach space $X$ is called approximatively polyhedral if for each $\varepsilon>0$ there is a polyhedral ($=$ intersection of finitely many closed half-spaces) convex set $P\subset X$ at Hausdorff distance $< \varepsilon$ from $C$. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component $\mathcal H$ of the space ${\rm Conv}_{\mathsf H}(X)$ of closed convex subsets of $X$ endowed with the Hausdorff metric is separable if and only if $\mathcal H$ contains a polyhedral convex set.
