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Bi-Lipschitz embeddings revisited

H. Movahedi-Lankarani, R. Wells

TL;DR

The paper investigates when the distance-to-point map $\iota_d:(X,d)\to L^2(\mu)$ can be used to obtain bi-Lipschitz embeddings of compact metric-measure spaces into finite-dimensional Euclidean spaces. It analyzes the canonical map, its radial projection, and associated metrics $\rho_d$ and $\theta_d$, establishing three concrete embedding criteria (via $\iota_{\rho_d}$, near-isometric behavior, and $\iota_{\Delta(d)}$) that guarantee a bi-Lipschitz embedding into some $\mathbb{R}^N$. Key contributions include three embedding theorems, a detailed examination of auxiliary maps and spherical compactness notions, and an erratum to a prior result (bilip) clarifying the required hypotheses. The results provide explicit, verifiable criteria for embedding compact metric-measure spaces into Euclidean spaces, with potential applications in geometric analysis and data representation, and connect canonical distance-function embeddings to broader concepts like Kał-doubling and spherical compactness.

Abstract

Given a metric space (X, d), we continue our study of the distance function x\mapsto d(x,-) and its relation to bi-Lipschitz embeddings of (X, d) into R^N. As application, given a compact metric-measure space (X, d,μ), we give three sufficient conditions for the existence of such a bi-Lipschitz embedding.

Bi-Lipschitz embeddings revisited

TL;DR

The paper investigates when the distance-to-point map can be used to obtain bi-Lipschitz embeddings of compact metric-measure spaces into finite-dimensional Euclidean spaces. It analyzes the canonical map, its radial projection, and associated metrics and , establishing three concrete embedding criteria (via , near-isometric behavior, and ) that guarantee a bi-Lipschitz embedding into some . Key contributions include three embedding theorems, a detailed examination of auxiliary maps and spherical compactness notions, and an erratum to a prior result (bilip) clarifying the required hypotheses. The results provide explicit, verifiable criteria for embedding compact metric-measure spaces into Euclidean spaces, with potential applications in geometric analysis and data representation, and connect canonical distance-function embeddings to broader concepts like Kał-doubling and spherical compactness.

Abstract

Given a metric space (X, d), we continue our study of the distance function x\mapsto d(x,-) and its relation to bi-Lipschitz embeddings of (X, d) into R^N. As application, given a compact metric-measure space (X, d,μ), we give three sufficient conditions for the existence of such a bi-Lipschitz embedding.
Paper Structure (7 sections, 18 theorems, 35 equations)

This paper contains 7 sections, 18 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The canonical map and radial projection
  3. Some related maps
  4. A bi-Lipschitz embedding result
  5. Another bi-Lipschitz embedding result
  6. A third bi-Lipschitz embedding result
  7. Remarks

Key Result

Lemma 2.1

Let $(X, d, \mu)$ be a compact metric-measure space with $\text{\rm{card}} (X) \ge 3$ and let $\iota_{d}: X \longrightarrow L^2 (\mu)$ be the canonical map. Then there exists $s > 0$ such that for all $x, y \in X$ we have $\langle\iota_{d} (x), \iota_{d} (y)\rangle \ge s > 0$.

Theorems & Definitions (27)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • ...and 17 more