When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?

TL;DR

This paper characterizes when the Gromov-Hausdorff distance between finite-dimensional Euclidean spaces and their subsets is finite, showing it occurs precisely when the subset is an epsilon-net, unlike in infinite dimensions.

Contribution

It provides a necessary and sufficient condition for finiteness of the Gromov-Hausdorff distance in finite-dimensional Euclidean spaces, extending understanding beyond infinite-dimensional cases.

Findings

Finite-dimensional Euclidean spaces have finite Gromov-Hausdorff distance to a subset iff the subset is an epsilon-net.

In infinite-dimensional spaces, this equivalence does not hold.

The proof uses upper estimates of Euclidean Gromov-Hausdorff distance.

Abstract

In this paper we prove that the Gromov--Hausdorff distance between $\mathbb{R}^n$ and its subset $A$ is finite if and only if $A$ is an $\varepsilon$-net in $\mathbb{R}^n$ for some $\varepsilon>0$. For infinite-dimensional Euclidean spaces this is not true. The proof is essentially based on upper estimate of the Euclidean Gromov--Hausdorff distance by means of the Gromov-Hausdorff distance.