On a Duality between Metrics and $\Sigma$-Proximities

TL;DR

This paper introduces a class of functions called $\Sigma$-proximities that are dual to metrics, measuring comparative proximity rather than distance, with applications to finite and infinite sets.

Contribution

It establishes a one-to-one correspondence between metrics and $\Sigma$-proximities, revealing their duality and properties in finite and infinite contexts.

Findings

$\Sigma$-proximities$ are dual to metrics.

They measure comparative proximity, not distance.

Diagonal entries indicate element centrality.

Abstract

: In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named $\Sigma$-proximities, are in a definite sense dual to metrics: there exists a natural one-to-one correspondence between metrics and $\Sigma$-proximities defined on the same finite set; in contrast to metrics, $\Sigma$-proximities measure {\it comparative} proximity; the closer the objects, the greater the $\Sigma$-proximity; diagonal entries of the $\Sigma$-proximity matrix characterize the ``centrality'' of elements. The results are extended to arbitrary infinite sets.