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

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty 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 = ∅}. 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)), thus extending the recent results due to Blasi, Myjak and Papini and Li.