New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912337771888640 |
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| author | Mikhailov, Ivan N. |
| author_facet | Mikhailov, Ivan N. |
| contents | In the paper we prove that, for arbitrary unbounded subset $A\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\times_{\ell^1} (tX)$, $t\in[0,\,\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $λ> 1$, $n\in\mathbb{N}$, the following inequality holds: $d_{GH}\bigl(\mathbb{Z}^n,\,λ\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. We conclude that a curve $t\mathbb{Z}^n$, $t\in(0,\,\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all metric spaces lying on the finite Gromov--Hausdorff distance from $\mathbb{R}^n$ on some~$λ> 0$ is also discontinous with respect to the Gromov--Hausdorff distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line Mikhailov, Ivan N. Metric Geometry 51F99, 53C22 In the paper we prove that, for arbitrary unbounded subset $A\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\times_{\ell^1} (tX)$, $t\in[0,\,\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $λ> 1$, $n\in\mathbb{N}$, the following inequality holds: $d_{GH}\bigl(\mathbb{Z}^n,\,λ\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. We conclude that a curve $t\mathbb{Z}^n$, $t\in(0,\,\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all metric spaces lying on the finite Gromov--Hausdorff distance from $\mathbb{R}^n$ on some~$λ> 0$ is also discontinous with respect to the Gromov--Hausdorff distance. |
| title | New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line |
| topic | Metric Geometry 51F99, 53C22 |
| url | https://arxiv.org/abs/2504.14629 |