Ambiguous loci of mutually nearest and mutually furthest points in Banach spaces (Q1879770)

This paper deals with the nature (in the topological sense) of the set where certain minimization or maximization problems are not well-posed. We recall that these problems are well-posed if there is a unique solution and every minimizing (maximizing) sequence converges strongly to the solution. Let \(X\) be a separable strictly convex Banach space. Let \({\mathcal K}(X)\) denote the family of non-empty boundedly compact convex sets equipped with the \(H_{\rho}\)-topology. Let \({\mathcal K}_{G}(X)\) denote the closure of the family of sets that are disjoint from \(G\). Let \({\mathcal V}(G) = \{A \in {\mathcal K}_{G}(X) : \min(A,G)\) is not well-posed\}. The main result of this paper shows that the collection of nonempty closed sets \(G\) for which \({\mathcal V}(G)\) is everywhere uncountable, is residual in the collection of all nonempty closed sets. Similar result is obtained in the case of farthest points by considering the Hausdorff distance and sets are assumed in addition to be bounded/compact.