Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness of best proximity pairs and rigidity of semimetric spaces   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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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: cond-mat