Ambiguous loci of mutually nearest and mutually furthest points in Banach spaces Original Research Article

Let X be a real separable strictly convex Banach space and G a nonempty closed subset of X. Let View the MathML source (resp. View the MathML source) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of X endowed with the Hρ-topology (resp. the Hausdorff distance), View the MathML source (resp. View the MathML source) the closure of the set View the MathML source (resp. View the MathML source), and View the MathML source (resp. View the MathML source) the family of View the MathML source (resp. View the MathML source) 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, View the MathML source (resp. View the MathML source) is everywhere uncountable in View the MathML source (resp. View the MathML source). A similar result for the mutually furthest point problem is also given.