Porosity of mutually nearest and mutually furthest points in Banach spaces

Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X : Let BðX Þ (resp. KðXÞ) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BGðXÞ denote the closure of the set fAABðX Þ : A-G 1⁄4 |g and KGðX Þ 1⁄4 BGðX Þ-KðXÞ: We introduce the admissible family A of BðX Þ and prove that E AðGÞ (resp. E o ðGÞ), the set of all subsets FAADBGðX Þ (resp. FAADBðX Þ) such that the minimization problem minðF ;GÞ (resp. the maximization problem maxðF ;GÞ) is well-posed, is a dense Gd-subset of A: Furthermore, when X is uniformly convex, we prove that A\E AðGÞ and A\E o ðGÞ are s-porous in A: r 2003 Elsevier Inc. All rights reserved. MSC: primary 41A65; 54E52; secondary 46B20