Metric Space Recognition by Gromov-Hausdorff Distances to Simplexes

TL;DR

This paper investigates how well Gromov-Hausdorff distances to simplexes can distinguish bounded metric spaces, providing conditions for when different spaces are non-distinguishable based on these distances.

Contribution

It introduces criteria for non-distinguishability of metric spaces using Gromov-Hausdorff distances to simplexes, advancing understanding of metric space recognition.

Findings

Examples of non-distinguishable metric spaces with non-zero Gromov-Hausdorff distance

Conditions for when metric spaces are non-distinguishable

Analysis of the limitations of simplexes in metric space recognition

Abstract

In the present paper a distinguishability of bounded metric spaces by the set of the Gromov--Hausdorff distances to so-called simplexes (metric spaces with unique non-zero distance) is investigated. It is easy to construct an example of non-isometric metric spaces such that the Gromov--Hausdorff distance between them vanishes. Such spaces are non-distinguishable, of course. But we give examples of non-distinguishable metric spaces, the Gromov--Hausdorff distance between which is non-zero. More over, we prove several sufficient and necessary conditions for metric spaces to be non-distinguishable.