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arXiv · 2510.05574

Locally similar distances and equality of the induced intrinsic distances

Erick Lee-Guzmán·Egor A. Maximenko·Enrique Abdeel Muñoz-de-la-Colina·Marco Iván Ruiz-Carmona
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Abstract

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.

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Erick Lee-Guzmán, Egor A. Maximenko, Enrique Abdeel Muñoz-de-la-Colina, Marco Iván Ruiz-Carmona. 2025-10-07. Locally similar distances and equality of the induced intrinsic distances. https://arxiv.org/abs/2510.05574

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