Porosity of mutually nearest and mutually furthest points in Banach spaces

AbstractLet X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X. Let B(X) (resp. K(X)) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BG(X) denote the closure of the set {A∈B(X):A∩G=∅} and KG(X)=BG(X)∩K(X). We introduce the admissible family A of B(X) and prove that EAo(G) (resp. EoA(G)), the set of all subsets F∈A⊆BG(X) (resp. F∈A⊆B(X)) such that the minimization problem min(F,G) (resp. the maximization problem max(F,G)) is well-posed, is a dense Gδ-subset of A. Furthermore, when X is uniformly convex, we prove that A⧹EAo(G) and A⧹EoA(G) are σ-porous in A