Porosity results for two-set nearest and farthest point problems

Abstract

Given two nonempty, closed and bounded subsets A and B of a complete geodesic space \((X,\rho ,M)\), we consider the problems of finding pairs of nearest and farthest points in A and B. Denoting by B(X) the family of all nonempty, closed and bounded subsets of X, we first endow \(B(X) \times B(X)\) with a pair of natural metrics. We then define corresponding metric spaces \({\mathcal {M}}\) of pairs (A, B) and construct subsets \(\Omega \) of \({\mathcal {M}}\) with \(\sigma \)-porous complements such that for each pair in \(\Omega \), these problems are well posed.