Asymptotic Chebyshev centers
The author studies properties of the set-valued map \(P_ M: B_{\infty}(X)\to M\cap B(X)\), where B(X) is the space of all nonempty bounded subsets of a normed linear space X with the Hausdorff metric h, \(B_{\infty}(X)\) is the metric space of all sequences \(\{C_ n\}\) of subsets \(C_ n\subset B(X)\) such that the union of \(C_ n\) is a bounded subset of X, endowed with the metric \(H(C,D)=\sup_{n}h(C_ n,D_ n),\) \(C=\{C_ n\}\), \(D=\{D_ n\}\in B_{\infty}(X)\) and \(P_ M(C)\), \(C=\{C_ m\}\in B_{\infty}(X)\) is the set of asymptotic Chebyshev centers of the sequence C with respect to a nonempty closed convex subset M of X. The author proves that asymptotic Chebyshev centers exist in a reflexive Banach space. The uniqueness of asymptotic Chebyshev centers are proved and a fixed-point theorem for Banach spaces that are uniformly convex in every direction is developed. Finally, the author shows that the map \(P_ M\) is uniformly and Hölder continuous on bounded subsets of \(B_{\infty}(X)\) in the case when the Banach space X is uniformly convex and q-convex, respectively.
scientific article; zbMATH DE number 4144652
scientific article