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arXiv · 2411.13539

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

I. N. Mikhailov·A. A. Tuzhilin
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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 $\varepsilon$-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.

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I. N. Mikhailov, A. A. Tuzhilin. 2024-11-20. When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?. https://arxiv.org/abs/2411.13539

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