Ambiguous loci of mutually nearest and mutually furthest points in Banach spaces

Let X be a real separable strictly convex Banach space and G a nonempty closed subset of X. Let K(X) (resp. K-b(X)) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of X endowed with the H-p-topology (resp. the Hausdorff distance), H-G(X) (resp. K-G(b)(X)) the closure of the set {A is an element of K(X) : A boolean AND G = 0} (resp. {A is an element of K-b(X) : A boolean AND G = 0}), and V(G) (resp. V-b(G)) the family of A is an element of K-G(X) (resp. A is an element of K-G(b)(X)) such that the minimization problem min(A, G) fails to be well-posed. It is proved that for most (in the sense of the Baire category) closed subsets (resp. bounded closed subsets) G of X, V(G) (resp. V-b(G)) is everywhere uncountable in K-G(X) (resp. K-G(b)(X)). A similar result for the mutually furthest point problem is also given. (C) 2004 Elsevier Ltd. All rights reserved.