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

### Tom 210 / 2012

Studia Mathematica 210 (2012), 137-157 MSC: Primary 52A07, 52A27; Secondary 46A55; 46N10, 52B05, 52A37. DOI: 10.4064/sm210-2-3

#### Streszczenie

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.

#### Autorzy

• Taras BanakhIvan Franko National University of Lviv
Lviv, Ukraine
and
Uniwersytet Jana Kochanowskiego
Kielce, Poland
e-mail
• Ivan HetmanIvan Franko National University of Lviv
Lviv, Ukraine
e-mail

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