On well-posed mutually nearest and mutually furthest point problems in Banach spaces
For a Banach space \(X\), denote by \(\mathcal B(X)\), \(\mathcal C(X)\), \(\mathcal K(X)\) the spaces of all nonempty closed bounded, nonempty closed bounded convex, respectively, nonempty compact convex subsets of \(X\). Equipped with the Hausdorff metric \(h\), \(\mathcal B(X)\) is a complete metric space and \(\mathcal C(X)\), \(\mathcal K(X)\) are closed subset of \(\mathcal B(X),\) hence complete, too. For a nonempty closed subset \(G\) of \(X\), denote by \(\mathcal K_G(X)\) the closure with respect to \(h\) of the set \(\{A\in \mathcal C(X) : A\cap G = \emptyset\}.\) The paper is concerned with the well-posedness of the problem \(\min(A,G)\) (resp. \(\max(A,G)\)) of finding \((x_0,z_0)\in A\times G\) such that \(\| z_0-x_0\| = \lambda_{AG}:= \inf\{\| z-x\| : x\in A\), \(z\in G\}\) (resp. \( \| z_0-x_0\| = \mu_{AG}:= \sup\{\| z-x\| : x\in A\), \(z\in G\}\)). The authors prove that if \(X\) is a strictly convex Banach space having the Kadec-Klee property and \(G\) is a nonempty closed boundedly relatively weakly compact subset of \(X,\) then 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 \(\mathcal K_ G(X)\) (Theorem 2.1). Under the same hypotheses on the space \(X\), if \(G\in \mathcal B(X)\) is relatively weakly compact, then the set of all \(A\in \mathcal K(X)\) for which the problem \(\max(A,G)\) is well posed contains a dense \(G_ \delta\) subset of \(\mathcal K(X)\) (Theorem 3.1). Similar results for \(X\) uniformly convex or strongly convex (meaning reflexive, strictly convex and Kadec-Klee) were obtained by \textit{F. S. De Blasi, J. Myjak} and \textit{P. L. Papini} [J. Approximation Theory 70, No. 2, 142--155 (1992; Zbl 0764.41032)], respectively by one of the authors, \textit{Chong Li} [J. Approximation Theory 103, No. 1, 1--17 (2000; Zbl 0952.46012)] and \textit{Chong Li} and \textit{Hong-Kun Xu} [Numer. Funct. Anal. Optimization 23, No. 3--4, 323--331 (2002; Zbl 1006.41024)].
scientific article; zbMATH DE number 2119279
scientific article