The Gromov-Hausdorff distance between $l^p$-products of metric spaces

TL;DR

This paper investigates the Gromov-Hausdorff distances between $l^p$-products of metric spaces, providing estimates, conditions for attainability, and specific examples including flat tori and spaces with their $l^$-product.

Contribution

It offers new estimates and conditions for the Gromov-Hausdorff distances between $l^p$-products, including explicit calculations for flat tori and spaces with large density.

Findings

Gromov-Hausdorff distance estimates for $l^p$-products

Conditions for the attainability of these estimates

Explicit calculation for flat tori and spaces with high density

Abstract

This paper studies $l^p$-products of metric spaces and provides estimates for the Gromov-Hausdorff distances between them. The case of linear products is considered separately, and sufficient conditions for attainability of the estimates are given for it. Examples of calculating the Gromov-Hausdorff distance between flat tori are given. It is proved that for any metric space $X$ of density $d(X)$, the Gromov-Hausdorff distance between it and its $l^\infty$-product (in which the number of factors corresponds to $d(X)$) is equal to half its diameter.