The max-type quasimetrics on probability simplices

TL;DR

This paper introduces a new class of quasimetrics on probability simplices inspired by Chebyshev distance, revealing their geometric properties and monotonicity under bistochastic maps, thus enriching the understanding of asymmetric distance measures.

Contribution

The paper presents a novel class of max-type quasimetrics on probability simplices, demonstrating their geometric structure and monotonicity properties, which were not previously known.

Findings

Quasimetrics induce Euclidean topology on simplices

They form geodesic spaces with Finslerian structure

They are monotone under bistochastic maps

Abstract

Quasimetric spaces form a natural framework to study distance problems with an inherent directional asymmetry. We introduce a simple novel class of quasimetrics on probability simplices, inspired by the Chebyshev distance. It is shown that such quasimetrics have expedient geometric properties -- they induce the Euclidean topology and a Finslerian infinitesimal structure, with which the probability simplices become geodesic spaces. Moreover, we prove that the broad family of the proposed quasimetrics are monotone under bistochastic maps.