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

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 $$d(x_1,\dots ,x_n)\le K\sum _{i=1}^{n}d(x_1,\dots ,x_n{)}_i^z,x_1,\dots ,x_n,z\in X,$$ where $K=1$. Here $d(x_1,\dots ,x_n{)}_i^z$ is obtained from the function $d(x_1,\dots ,x_n)$ 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.

References in corpus (1)

• Minsum Location Extended to Gauges and to Convex Sets

Cited by in corpus (4)

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