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

Abstract

For a Banach space $X$ by $Conv_H(X)$ we denote the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric. We prove that for any closed convex set $C\subset X$ and its metric component $H_C=\{A\in Conv_H(X):d_H(A,C)<\infty\}$ in $Conv_H(X)$, the following conditions are equivalent: (1) $C$ is approximatively polyhedral, which means that for every $\epsilon>0$ there is a polyhedral convex subset $P\subset X$ on Hausdorff distance $d_H(P,C)<\epsilon$ from $C$; (2) $C$ lies on finite Hausdorff distance $d_H(C,P)$ from some polyhedral convex set $P\subset X$; (3) the metric space $(H_C,d_H)$ is separable; (4) $H_C$ has density $dens(H_C)<\mathfrak c$; (5) $H_C$ does not contain a positively hiding convex set $P\subset X$. If the Banach space $X$ is finite-dimensional, then the conditions (1)--(5) are equivalent to: (6) $C$ is not positively hiding; (7) $C$ is not infinitely hiding. A convex subset $C\subset X$ is called {\em positively hiding} (resp. {\em infinitely hiding}) if there is an infinite set $A\subset X\setminus C$ such that $\inf_{a\in A}dist(a,C)>0$ (resp. $\sup_{a\in A}dist(a,C)=\infty$) and for any distinct points $a,b\in A$ the segment $[a,b]$ meets the set $C$.