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

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 $\epsilon$-net in ${\mathbb{R}}^n$ for some $\epsilon >0$. 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.

6 pages, accepted for publication in Commun. Math. Res. (http://www.global-sci.org/intro/online.html?journal=cmr)