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arXiv · 1611.07826

A generalization of the concept of distance based on the simplex inequality

Gergely Kiss·Jean-Luc Marichal·Bruno Teheux
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Abstract

We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, $$ where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ for which the inequality above holds. We also introduce a generalization of the concept of $n$-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function.

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Gergely Kiss, Jean-Luc Marichal, Bruno Teheux. 2018-05-22. A generalization of the concept of distance based on the simplex inequality. https://doi.org/10.1007/s13366-018-0379-5

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