最佳邻近对的唯一性与半度量空间的刚性
Uniqueness of best proximity pairs and rigidity of semimetric spaces
摘要：
对于任意半度量空间$(X, d)$和$X$的不交邻近子集$A$，$B$，我们定义邻近图为具有部分$A$和$B$的二部图，其边$\{a, b\}$满足等式$d(a, b) = \operatorname{dist}(A, B)$ .刻画了邻近图至多有一条边的半度量空间和邻近图的顶点至多只有$1$的半度量空间。这使我们能够描述的充要条件的唯一性的最佳邻近对和最佳逼近。
英文摘要：
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.
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