Calculating Gromov-Hausdorff distance by means of asymptotic dimension

TL;DR

This paper explores how the concept of asymptotic dimension can be used to compute Gromov-Hausdorff distances between unbounded metric spaces, providing explicit calculations for specific examples.

Contribution

It introduces a novel approach linking asymptotic dimension to Gromov-Hausdorff distance calculations for unbounded metric spaces.

Findings

Gromov-Hausdorff distance between  and  equals their Hausdorff distance.

Application of asymptotic dimension simplifies Gromov-Hausdorff distance calculations.

Explicit example with  and  demonstrates the method.

Abstract

In this paper, we apply the concept of asymptotic dimension to calculating Gromov-Hausdorff distances between some unbounded metric spaces. For example, we show that the Gromov--Hausdorff between $\mathbb{R}^2$ with the Euclidean metric and $\mathbb{Z}^2$ equals the Hausdorff distance between them: $d_{GH}(\mathbb{R}^2, \mathbb{Z}^2) = d_H(\mathbb{R}^2, \mathbb{Z}^2)$.