Mutually nearest and mutually farthest points of sets in geodesic spaces

TL;DR

This paper investigates the well-posedness of mutual nearest and farthest point problems in geodesic spaces, establishing generic conditions and specific results for convex, compact, and CAT(0) spaces, with applications to optimization.

Contribution

It provides new generic well-posedness results for extremal point problems in geodesic spaces, including spaces with curvature bounds and convex metrics, extending classical results to broader settings.

Findings

Well-posedness is dense in the space of convex, closed, bounded sets.

Results apply to CAT(0) spaces and include a variant of the Drop theorem.

Conditions for well-posedness depend on space geometry and set properties.

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.