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

Kiss, Gergely and Marichal, Jean-Luc and Teheux, Bruno (2018) A generalization of the concept of distance based on the simplex inequality. BEITRÄGE ZUR ALGEBRA UND GEOMETRIE, 59 (2). pp. 247-266. ISSN 0138-4821

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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 d(x1,…,xn)≤K∑i=1nd(x1,…,xn)zi,x1,…,xn,z∈X, where K=1. Here d(x1,…,xn)zi is obtained from the function d(x1,…,xn) 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∈]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.

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