Porosity of mutually nearest and mutually furthest points in Banach spaces.
In this paper the authors study well-posedness of certain minimization and maximazation problems in strictly convex Banach spaces \(X\) with the Kadec property (i.e, weak and norm sequential convergence coincide on the unit sphere). Let \({\mathcal B}(X)\) be the set of non-empty closed and bounded subsets of \(X\) equipped with the Hausdorff metric. A closed subset \({\mathcal A }\subset {\mathcal B}(X)\) is said to be an admissible family if \(A \cup \{x\} \in \mathcal A\) whenever \(x \in X\) and \(A \in {\mathcal A}\). Let \(G \subset X\) be a non-empty relatively weakly compact and bounded set. One of the results proved in this paper states that \(\{F \in {\mathcal A}\): the maximization problem \(\max(F,G)\) is well-posed\} is a dense \(G_{\delta}\) subset of \({\mathcal A}\). A similar result is proved for the minimization problem. We recall that for a metric space \((E,d)\), \(Y \subset E\) is said to be porous if there exists \(0<t \leq 1\) and \(r_0 >0\) such that for every \(x \in E\) and \(0<r \leq r_0\) there is a \(y \in E\) such that \(B_d(y,tr) \subset B_d(x,r) \cap (E\setminus Y)\). Under the assumption of uniform convexity on \(X\) it is also proved that the set \({\mathcal A} \setminus \{F \in {\mathcal B}(X) \): the minimization problem \(\min (F,G)\) is well-posed\} is \(\sigma\)-porous in \({\mathcal A}\).
Mutually nearest and mutually farthest points
scientific article; zbMATH DE number 2026867