ON MUTUALLY NEAREST AND MUTUALLY FURTHEST POINTS OF SETS IN BANACH-SPACES

Let A be a nonempty closed bounded subset of a uniformly convex Banach space E. Let b(E) denote the space of all nonempty closed convex and bounded subsets of E, endowed with the Hausdorff metric. We prove that the set of all X ε{lunate} E(E) such that the maximization problem max(A, X) is well posed is a Gδ dense subset of b(E). A similar result is proved for the minimization problem min(A, X), with X in an appropriate subspace of b(E). © 1992.