Locally similar distances and equality of the induced intrinsic distances
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arXiv
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| Format: | Preprint |
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2025
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| author | Lee-Guzmán, Erick Maximenko, Egor A. Muñoz-de-la-Colina, Enrique Abdeel Ruiz-Carmona, Marco Iván |
| author_facet | Lee-Guzmán, Erick Maximenko, Egor A. Muñoz-de-la-Colina, Enrique Abdeel Ruiz-Carmona, Marco Iván |
| contents | Let $X$ be a set and $d_1,d_2$ be two distances on $X$. We say that $d_1$ and $d_2$ are locally similar and write $d_1\cong d_2$ if $d_1$ and $d_2$ are topologically equivalent and, for every $a$ in $X$, \[ \lim_{x\to a} \frac{d_2(x,a)}{d_1(x,a)}=1. \] We prove that if $d_1\cong d_2$, then the intrinsic distances induced by $d_1$ and $d_2$ coincide. We also provide sufficient conditions for $d_1\cong d_2$ and consider several examples related to reproducing kernel Hilbert spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally similar distances and equality of the induced intrinsic distances Lee-Guzmán, Erick Maximenko, Egor A. Muñoz-de-la-Colina, Enrique Abdeel Ruiz-Carmona, Marco Iván Metric Geometry Functional Analysis 51K05, 30F45, 46E22 Let $X$ be a set and $d_1,d_2$ be two distances on $X$. We say that $d_1$ and $d_2$ are locally similar and write $d_1\cong d_2$ if $d_1$ and $d_2$ are topologically equivalent and, for every $a$ in $X$, \[ \lim_{x\to a} \frac{d_2(x,a)}{d_1(x,a)}=1. \] We prove that if $d_1\cong d_2$, then the intrinsic distances induced by $d_1$ and $d_2$ coincide. We also provide sufficient conditions for $d_1\cong d_2$ and consider several examples related to reproducing kernel Hilbert spaces. |
| title | Locally similar distances and equality of the induced intrinsic distances |
| topic | Metric Geometry Functional Analysis 51K05, 30F45, 46E22 |
| url | https://arxiv.org/abs/2510.05574 |