Mutually nearest and mutually farthest points of sets in geodesic spaces

Details Mutually nearest and mutually farthest points of sets in geodesic spaces Mutually nearest and mutually farthest points of sets in geodesic spaces

2010-03-22

arxiv.org/abs/1003.4179v1

Mathematics

Metric Geometry

20 pages, research paper

Scientific paper

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.

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