Uniqueness of best proximity pairs and rigidity of semimetric spaces

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 $d(a,b)=\mathrm{dist}(A,B)$. 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.

32 pages, 10 figures