Mutually nearest and mutually farthest points of sets in geodesic spaces

Abstract

Let $A$ and $X$ be nonempty, bounded and closed subsets of a geodesic metric space $(E,d)$. The minimization (resp. maximization) problem denoted by $\min (A,X)$ (resp. $\max (A,X)$) consists in finding $(a_0,x_0)\in A\times X$ such that $d(a_0,x_0)=\inf \{d(a,x):a\in A,x\in X\}$ (resp. $d(a_0,x_0)=\sup \{d(a,x):a\in A,x\in X\}$). We study the well-posedness of these problems in different geodesic spaces considering the set $A$ fixed. Let $P_{b,cl,cv}(E)$ be the space of all nonempty, bounded, closed and convex subsets of $E$ endowed with the Pompeiu-Hausdorff distance. We show that in a space with a convex metric, curvature bounded below and the geodesic extension property, the family of sets in $P_{b,cl,cv}(E)$ for which $\max (A,X)$ is well-posed is a dense $G_{\delta }$-set in $P_{b,cl,cv}(E)$. We give a similar result for $\min (A,X)$ without needing the geodesic extension property. Besides, we analyze the situations when one set or both sets are compact and prove some results specific to CAT$(0)$ spaces. We also prove a variant of the Drop theorem in geodesic spaces with a convex metric and apply it to obtain an optimization result for convex functions.

20 pages, research paper