Porosity of Mutually Nearest and Mutually Furthest Points Problems in Geodesic Space

Abstract

Let X be a nonempty, bounded, and closed subset of a geodesic metric space (E, d). Let 𝔅 X (E) denote the family of nonempty, bounded, closed subsets of E endowed with the Pompeiu-Hausdorff distance and let 𝔅 X (E) denote the closure of the set {A ∈ 𝔅(E): A ∩ X = ∅}. The mutually nearest (resp., furthest) problem considered here is to find (a 0; x 0) with a 0 ∈ A; x 0 ∈ X such that d(a 0; x 0) = inf {d(a; x): a ∈ A; x ∈ X} (resp., d(a 0; x 0) = sup {d(a; x): a ∈ A; x ∈ X}). Let 𝔄 ⊆ 𝔅 X (E) be the admissible family. We show that 𝒲min, the set of all subsets A ∈ 𝔅 X (E) such that the mutually nearest point problem min (A; X) is well posed, is a dense Gδ-subset of 𝔄. In fact we show more, namely that 𝔄∖𝒲min is σ-porous in 𝔄. Moreover, we prove similar results for the mutually furthest point problem. In particular, our works generalize and sharpen some results from [13, 17, 19].