On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Abstract

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let $$ {\user1{K}}{\left( X \right)} $$ denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let $$ {\user1{K}}_{G} {\left( X \right)} $$ denote the closure of the set $$ {\left\{ {A \in {\user1{K}}{\left( X \right)}:A \cap G = \emptyset } \right\}} $$ . We prove that the set of all $$ A \in {\user1{K}}_{G} {\left( X \right)}{\left( {{\text{resp}}{\text{.}}{\kern 1pt} A \in {\user1{K}}{\left( X \right)}} \right)} $$ , such that the minimization (resp. maximization) problem min(A,G) (resp. max(A,G)) is well posed, contains a dense G δ-subset of $$ {\user1{K}}_{G} {\left( X \right)}{\left( {{\text{resp}}{\text{.}}{\kern 1pt} {\kern 1pt} {\user1{K}}{\left( X \right)}} \right)} $$ , thus extending the recent results due to Blasi, Myjak and Papini and Li.