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

TL;DR

This paper introduces the concept of n-distance, a generalization of classical distance based on the simplex inequality, explores various examples, and investigates the bounds of the inequality's constant.

Contribution

It proposes a new generalized distance concept called n-distance, extending the triangle inequality to the simplex inequality, with examples and analysis of the inequality bounds.

Findings

Several examples of n-distances are provided.

The infimum of the constant K for which the inequality holds is investigated.

A generalized n-distance using arbitrary symmetric functions is introduced.

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.