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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

TL;DR

Addresses when topological and pointwise local similarity of distances, expressed as $d_1\\cong d_2$, ensures equality of their intrinsic distances $d^*$ and $d^*$. The authors introduce locally similar distances, prove the main result $d_1^*=d_2^*$, and develop sufficient conditions for $d_2=f\\circ d_1$, including strong and infinitesimal variants. They provide kernel-centered examples showing how $d^*$ can be computed from base distances via concave transforms, with explicit constants in RKHS contexts. The results connect elementary metric notions with kernel-induced metrics and yield practical tools for intrinsic-distance computations in analysis and geometry.

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 $d_2$ are equal. We also provide some sufficient conditions for $d_1\cong d_2$ and consider several examples related to reproducing kernel Hilbert spaces.

Locally similar distances and equality of the induced intrinsic distances

TL;DR

Addresses when topological and pointwise local similarity of distances, expressed as , ensures equality of their intrinsic distances and . The authors introduce locally similar distances, prove the main result , and develop sufficient conditions for , including strong and infinitesimal variants. They provide kernel-centered examples showing how can be computed from base distances via concave transforms, with explicit constants in RKHS contexts. The results connect elementary metric notions with kernel-induced metrics and yield practical tools for intrinsic-distance computations in analysis and geometry.

Abstract

Let be a set and be two distances on . We say that and are locally similar and write , if and are topologically equivalent and for every in , We prove that if , then the intrinsic distances induced by and are equal. We also provide some sufficient conditions for and consider several examples related to reproducing kernel Hilbert spaces.

Paper Structure

This paper contains 7 sections, 17 theorems, 148 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The intrinsic distance induced by a given distance
  3. Locally similar distances
  4. Strongly locally similar and infinitesimally similar distances
  5. Locally similar distances and composition
  6. Locally similar distances induce the same intrinsic distance
  7. Examples related to kernels

Key Result

Lemma 2.4

Let $(X, d)$ be a metric space, $x, y\in X$, and $\gamma\in\Gamma_d(x, y)$. If $P, Q\in\mathcal{P}$ such that $P\subseteq Q$, then

Figures (5)

  • Figure 1: Some logic relations between different equivalencies of distances.
  • Figure 2: Distances from Example \ref{['example:open_interval_with_circular_distance']}.
  • Figure 3: Part of the comb from Example \ref{['example:comb']}. The light-green thick lines illustrate the shortest path from $(0,0)$ to $\left(\frac{1}{2},\frac{1}{2}\right)$.
  • Figure 4: Distances from Example \ref{['example:hook']}.
  • Figure 5: Lemma \ref{['lem:interval_inclusion']}, case (a) (above) and (b) (below).

Theorems & Definitions (68)

  • Definition 2.1: continuous paths between two points
  • Definition 2.2: Riemannian partitions of the unit interval
  • Definition 2.3: the polygonal length of a path with respect to a partition
  • Lemma 2.4
  • proof
  • Definition 2.5: the length of a path with respect to a distance
  • Definition 2.6: the intrinsic distance induced by a distance
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 58 more