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Volume 41, Issue 1
When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?

I.N. Mikhailov & A.A. Tuzhilin

DOI: 10.4208/cmr.2024-0041

Commun. Math. Res., 41 (2025), pp. 1-8.

Published online: 2025-03

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

  • Keywords

Metric space, $ε$-net, Gromov-Hausdorff distance.

  • AMS Subject Headings

46B20, 51F99

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{CMR-41-1, author = {Mikhailov , I.N. and Tuzhilin , A.A.}, title = {When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?}, journal = {Communications in Mathematical Research }, year = {2025}, volume = {41}, number = {1}, pages = {1--8}, 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 $ε$-net in $\mathbb{R}^n$ for some $ε > 0.$ 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2024-0041}, url = {http://global-sci.org/intro/article_detail/cmr/23926.html} }
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 -

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 $ε > 0.$ 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.

Mikhailov , I.N. and Tuzhilin , A.A.. (2025). When the Gromov-Hausdorff Distance Between Finite-Dimensional Space and Its Subset Is Finite?. Communications in Mathematical Research . 41 (1). 1-8. doi:10.4208/cmr.2024-0041
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