Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance

TL;DR

This paper explores the universality of the Gromov-Hausdorff distance among orbit pseudometrics, establishing a bireducibility result that links it to a universal orbit pseudometric in the context of Borel reducibility.

Contribution

It extends the concept of universality from orbit equivalence relations to orbit pseudometrics, specifically demonstrating the Gromov-Hausdorff distance's universality.

Findings

Gromov-Hausdorff distance is bireducible with a universal orbit pseudometric.

Establishes a connection between Borel reducibility and orbit pseudometrics.

Provides a new perspective on the complexity of metric space classification.

Abstract

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by C\'{u}th, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of an orbit equivalence relation. It is well known that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. We prove a version of this result for pseudometrics, showing that the Gromov-Hausdorff distance of Polish metric spaces is bireducible with a universal element in a certain class of orbit pseudometrics.