On mutually nearest and mutually farthest points in reflexive Banach spaces
Let \(X\) be a real Banach space which is reflexive, strictly convex, and with a Kadets norm (if (\(x_n)\) converges to \(x\) weakly and \((\|x_n\|)\) converges to \(\|x\|\) them \((\|x_n-x\|)\) converges to 0). Let \({\mathcal K}(X)\) be the set of the nonempty compact convex subsets of \(X\), endowed with the Hausdorff metric, and, for \(G\subseteq X\) nonempty and closed, let \({\mathcal K}_G(X)\) be the closure of \(\{A\in{\mathcal K}(X) ;\;A\cap G=\emptyset\}\). It is said that the problem \(\min(A,G)\) is ``well posed'' if there is a unique \((x_0,y_0)\in A\times G\) such that \(\|x_0-y_0\|=\inf\{\|x-y\|;\;x\in A,\;y\in G\}\) and every minimizing sequence \(\{(x_n,y_n)\}\) converges strongly to \((x_0,y_0)\). The author shows that the set of all \(A\in{\mathcal K}_G(X)\) for which the problem \(\min(A,G)\) is well posed contains a dense \(G_\delta\) subset of \({\mathcal K}_G(X)\). The denseness depends on a difficult result by \textit{D. Preiss} about differentiability of scalar Lipschitz functions on Banach spaces [J. Funct. Anal. 91, No. 2, 312-345 (1990; Zbl 0711.46036)]. In a second part, the author shows an analogous result for the maximization problem \(\sup\{\|x-y\|;\;x\in A,\;y\in G\}\) (for bounded \(G\)).
scientific article; zbMATH DE number 1423070
scientific article