Uniqueness of best proximity pairs and rigidity of semimetric spaces

TL;DR

This paper characterizes semimetric spaces based on the structure of their proximinal graphs, providing conditions for the uniqueness of best proximity pairs and approximations.

Contribution

It introduces a characterization of semimetric spaces with unique or limited proximinal graph edges, advancing understanding of best proximity pairs.

Findings

Semimetric spaces with at most one proximinal graph edge are characterized.

Conditions for vertices with degree at most one in proximinal graphs are established.

Necessary and sufficient conditions for uniqueness of best proximity pairs are provided.

Abstract

For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most $1$ only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.