A ``hidden'' characterization of approximatively polyhedral convex sets in Banach spaces (Q2907266)

Let \(X\) be a Banach space. A convex set \(C\) in \(X\) is called a closed half-space if \(C = f^{-1}[a,\infty)\) for some non-zero linear continuous functional on \(X\). It is called polyhedral if \(C\) can be written as the intersection of a finite number of closed half-spaces. It is called approximately polyhedral if for each \(\epsilon > 0\), there exists a closed polyhedral subset \(D\) such that \(d_H(C,D)< \epsilon\) where \(d_H(C,D)\) is the Hausdorff metric between \(C\) and \(D\).NEWLINENEWLINELet \(Conv_H(X)\) be the space of all closed convex subsets of \(X\) endowed with the Hausdorff metric \(d_H(A,B)\). For \(C\in Conv_H(X)\) the Hausdorff distance \(d_H\) restricted to \(H_C =\{A\in Conv_H(X) : d_H(A,C)< \infty\}\) is a metric. The resulting metric space \((H_C,d_H)\) is called the component of \(C\) in \(Conv_H(X)\). A subset \(C\) of a linear space \(E\) hides a set \(A \subset E\) if for any distinct points \(a, b\) in \(A\), \([a, b] = \{ta +(1-t)b :t\in [0, 1]\}\) meets \(C\). A convex subset \(C\) in \(X\) is hiding if \(C\) hides some infinite set \(A\subset X/C\). \(C\) is called positively hiding if \(C\) hides some infinite set \(A\subset X/C\) such that \(sup_{a\in A}dist(a,C) > 0\). \(C\) is called infinitely hiding if \(C\) hides some infinite set \(A \subset X/C\) such that \(sup_{a\in A}dist(a,C) = \infty \). Let us recall that the density \(dens(X)\) of a topological space \(X\) is the smallest cardinality of a dense subset of \(X\). Topological spaces with at most countable density are called separable.NEWLINENEWLINEThe main result of the paper is the following characterization theorem.NEWLINENEWLINE{ Theorem.} For a closed convex subset \(C\) of a Banach space \(X\) the following are equivalent: {\parindent=0.5cm\begin{itemize}\item[1)] \(C\) is approximately polyhedral. \item[2)] The characteristic cone \(V_C\) is polyhedral in \(X\) and \(h_H(C,V_C)< \infty\) \item[3)] The component \( H_C\) contains a polyhedral closed convex set, \item[4)] \({H_C}\) contains no positively hiding closed convex set, \item[5)] \( H_C\) is separable, \item[6)] \( dens( H_C)< |\Re|\).NEWLINENEWLINEIf \(X\) is finite dimensional then (1)-(6) are equivalent to : \item[7)] \(C\) is not positively hiding, \item[8)] \(C\) is infinitely hiding.NEWLINENEWLINE\end{itemize}}