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 of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex set.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-2-3,
author = {Taras Banakh and Ivan Hetman},
title = {A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {137-157},
zbl = {1268.52001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-2-3}
}
Taras Banakh; Ivan Hetman. A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces. Studia Mathematica, Tome 209 (2012) pp. 137-157. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-2-3/