The max-type quasimetrics on probability simplices

2025/10/31 by Michał Eckstein, Eckstein, Michał, Tomasz Miller +3
Mathematics · Physics and Astronomy · #51F99 #53B12 #53B40 #Advanced Differential Geometry Research #FOS: Mathematics #FOS: Physical sciences #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2510.27368

openalex publication_date 2025/10/31 · openalex created_date 2025/11/05 · openalex updated_date 2026/07/28

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.

Citations

• Impact of one-way streets on the asymmetry of the shortest commuting routes
• Combining Minkowski and Chebyshev: New distance proposal and survey of distance metrics using k-nearest neighbours classifier
• Quasi-metric and metric spaces
• On an Asymmetrical Metric in the Four-Space of General Relativity

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• Geodesics for a class of distances in the space of probability measures
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