The geodesic problem in quasimetric spaces

TL;DR

This paper investigates geodesics in quasimetric spaces, generalizing metric space results, and shows how certain quasimetrics induce intrinsic metrics where optimal transport paths are geodesics.

Contribution

It extends classical metric space results to quasimetric spaces and introduces intrinsic metrics where optimal transport paths are geodesics.

Findings

Many metric space theorems hold in quasimetric spaces

Conditions for quasimetrics to induce intrinsic metrics are identified

Optimal transport paths are geodesics in these intrinsic metric spaces

Abstract

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq \sigma (d(x,z)+d(z,y))$ for some constant $\sigma \geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzel\`{a} theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_{\alpha}$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a "tree shaped" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.