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=0}．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．