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

paper

Locally similar distances and equality of the induced intrinsic distances

arXiv:2510.05574

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 $d_{1} ≅ d_{2}$ 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 $d_{1} ≅ d_{2}$ , then the intrinsic distances induced by $d_{1}$ and $d_{2}$ coincide. We also provide sufficient conditions for $d_{1} ≅ d_{2}$ and consider several examples related to reproducing kernel Hilbert spaces.

32 pages, 6 figures. In the second version, we have changed the order of sections and added more examples