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

Let <f>X</f> be a real separable strictly convex Banach space and <f>G</f> a nonempty closed subset of <f>X</f>. Let <f>K(X)</f> (resp. <f>Kb(X)</f>) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of <f>X</f> endowed with the <f>Hρ</f>-topology (resp. the Hausdorff distance), <f>KG(X)</f> (resp. <f>KGb(X)</f>) the closure of the set <f>{A∈K(X): A∩G=}</f> (resp. <f>{A∈Kb(X): A∩G=}</f>), and <f>V(G)</f> (resp. <f>Vb(G)</f>) the family of <f>A∈KG(X)</f> (resp. <f>A∈KGb(X)</f>) such that the minimization problem <f>min(A,G)</f> fails to be well-posed. It is proved that for most (in the sense of the Baire category) closed subsets (resp. bounded closed subsets) <f>G</f> of <f>X</f>, <f>V(G)</f> (resp. <f>Vb(G)</f>) is everywhere uncountable in <f>KG(X)</f> (resp. <f>KGb(X)</f>). A similar result for the mutually furthest point problem is also given. [Copyright &y& Elsevier]