Chebyshev centers and some geometric properties of Banach spaces
Let \(X\) be a real Banach space, \(\emptyset\neq V\subset X\) closed and \(\emptyset\neq F\subset X\) bounded. For \(v\in V\) and \(\delta >0\) put \(r(v,F)=\sup\{\|v-x\| : x\in F\}\), \(\mathrm{rad}_V(F)=\inf\{r(v,F) : v\in V\}\) (the Chebyshev radius of \(F\) with respect to \(V\)), \(\mathrm{cent}_V(F)=\{v\in V : v(v,F)= \mathrm{rad}_V(F)\}\) (the Chebyshev center of \(F\) with respect to \(V\)) and \(\delta\)-\(\mathrm{cent}_V(F)=\{v\in V : v(v,F)\leq \mathrm{rad}_V(F)+\delta\}\). Denote also by \(CL(X), CB(X), CC(X)\) and \(K(X)\) the family of all nonempty closed subsets of \(X\), nonempty closed bounded subsets of \(X\), nonempty closed convex subsets of \(X\), and nonempty compact subsets of \(X\), respectively. Let \(V\in CL(X)\) and \(\mathcal F\subset CB(X)\) be such that \(\mathrm{cent}_V(F)\neq\emptyset\) for every \(F\in\mathcal F\). One says that the pair \((V,\mathcal F)\) has the property: {\parindent=9mm \begin{itemize}\item[(P1)] if, for every \(F\in\mathcal F\) and \(\varepsilon >0\), there exists \(\delta>0\) such that \(\delta\)-\(\mathrm{cent}_V(F)\subset \mathrm{cent}_V(F)+\varepsilon B_X\); \item[(P2)] if, for every \(\varepsilon >0\), there exists \(\delta>0\) such that \(\delta\)-\(\mathrm{cent}_V(F)\subset \mathrm{cent}_V(F)+\varepsilon B_X\) for all \(F\in\mathcal F\). \end{itemize}} The properties (P1) and (P2) were introduced by \textit{J. Mach} [J. Approx. Theory 29, 223--230 (1980; Zbl 0467.41015)] in his study of the continuity properties of the Chebyshev center map. The main aim of the paper is the study of properties (P1) and (P2) in connection with some geometric properties of Banach spaces. For example: \(X\) is reflexive with the Kadets-Klee property iff the pair \((V,K(X))\) has property (P1) for every \(V\in CC(X)\) (Theorem 2.3). If \(X\) is compactly locally uniformly convex (CLUR), then, for fixed \(V\in CC(X)\), the pair \((V,\mathcal F)\) has property (P1), where \(\mathcal F=\{F\in K(X) : \mathrm{cent}_V(F)\neq\emptyset\}\) (Theorem 2.5). Also, the \(\delta\)-center map, \(\delta\)-\(\mathrm{cent}_V:CCB(X)\to CCB(X)\), is uniformly continuous on bounded sets with respect to the Hausdorff metric for every \(V\in CC(X)\) and \(\delta>0\) (Theorem 2.10). Another result, Theorem 3.1, asserts that, if the pair \((X,\mathcal F) \) has property (P2), where \(V\in CL(X)\) and \(\mathcal F\subset CB(X)\), then the center map \(\mathrm{cent}_V:\mathcal F\to CCB(X)\) is uniformly continuous with respect to the Hausdorff metric. The map \(F\mapsto\mathrm{cent}_X(F)\) is single-valued and uniformly continuous on bounded sets with respect to the Hausdorff metric iff the Banach space \(X\) is uniformly convex (Theorem 3.8).
scientific article; zbMATH DE number 6675628