A “hidden” characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh¹ ; Ivan Hetman²

¹ Ivan Franko National University of Lviv Lviv, Ukraine and Uniwersytet Jana Kochanowskiego Kielce, Poland
² Ivan Franko National University of Lviv Lviv, Ukraine

Studia Mathematica, Tome 210 (2012) no. 2, pp. 137-157

Résumé

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.

Détail

DOI : 10.4064/sm210-2-3

Keywords: closed convex subset banach space called approximatively polyhedral each varepsilon there polyhedral intersection finitely many closed half spaces convex set subset hausdorff distance varepsilon nbsp characterize approximatively polyhedral convex sets banach spaces apply characterization connected component mathcal space conv mathsf closed convex subsets endowed hausdorff metric separable only mathcal contains polyhedral convex set