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

Abstract

We introduce and discuss the concept of 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

$$\begin{aligned} d(x_1, \ldots , x_n)\le K\, \sum _{i=1}^n d(x_1, \ldots , x_n)_i^z, \quad x_1, \ldots , x_n, z \in X, \end{aligned}$$

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 ith 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 $K\in \,]0,1]$ 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.