On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces Original Research Article

Let G be a nonempty closed (resp. bounded closed) subset in a reflexive strictly convex Kadec Banach space X. Let K(X) denote the space of all nonempty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A∈K(X) : A∩G=∅}. A minimization problem min(A, G) (resp. maximization problem max(A, G)) is said to be well posed if it has a unique solution (x0, z0) and every minimizing (resp. maximizing) sequence converges strongly to (x0, z0). We prove that the set of all A∈KG(X) (resp. A∈K(X)) such that the minimization (resp. maximization) problem min(A, G) (resp. max(A, G)) is well posed contains a dense Gδ-subset of KG(X) (resp. K(X)), extending the results in uniformly convex Banach spaces due to Blasi, Myjak and Papini.