TY - JOUR
T1 - When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?
AU - Mikhailov , I.N.
AU - Tuzhilin , A.A.
JO - Communications in Mathematical Research 
VL - 1
SP - 1
EP - 8
PY - 2025
DA - 2025/03
SN - 41
DO - http://doi.org/10.4208/cmr.2024-0041
UR - https://global-sci.org/intro/article_detail/cmr/23926.html
KW - Metric space, $ε$-net, Gromov-Hausdorff distance.
AB - <p style="text-align: justify;">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 $ε$-net in $\mathbb{R}^n$ for some $ε &gt; 0.$ 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.</p>