When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Mikhailov, I. N. Tuzhilin, A. A. |
| author_facet | Mikhailov, I. N. Tuzhilin, A. A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite? Mikhailov, I. N. Tuzhilin, A. A. Metric Geometry 46B20, 51F99 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. |
| title | When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite? |
| topic | Metric Geometry 46B20, 51F99 |
| url | https://arxiv.org/abs/2411.13539 |