# On the best constants associated with $n$-distances

## TL;DR

This paper explores the concept of $n$-distances, focusing on computing optimal constants for inequalities, defining subclasses, and linking $n$-distances to multidistances, advancing theoretical understanding of these generalized metrics.

## Contribution

It introduces methods to compute best constants for $n$-distances, defines subclasses based on properties, and establishes connections with multidistances, expanding the theoretical framework.

## Key findings

- Computed best constants for partial simplex inequalities.
- Defined subclasses of $n$-distances based on properties.
- Established links between $n$-distances and multidistances.

## Abstract

We pursue the investigation of the concept of $n$-distance, an $n$-variable version of the classical concept of distance recently introduced and investigated by Kiss, Marichal, and Teheux. We especially focus on the challenging problem of computing the best constant associated with a given $n$-distance. In particular, we define and investigate the best constants related to partial simplex inequalities. We also introduce and discuss some subclasses of $n$-distances defined by considering some properties. Finally, we discuss an interesting link between the concepts of $n$-distance and multidistance.