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Nonlinear type and metric embeddings of lamplighter spaces

C. Gartland, B. Randrianantoanina, N. L. Randrianarivony

Abstract

We prove that for all metric spaces $X$ the following properties of the lamplighter space $\mathsf{La}(X)$ are equivalent: (1) $\mathsf{La}(X)$ has finite Nagata dimension, (2) $\mathsf{La}(X)$ has Markov type 2, (3) $\mathsf{La}(X)$ does not contain the Hamming cubes with uniformly bounded biLipschitz distortion, (4) $\mathsf{La}(X)$ admits a weak biLipschitz embedding into a finite product of $\mathbb{R}$-trees. We characterize metric spaces $X$ for which $\mathsf{La}(X)$ satisfies properties (1)-(4) as those whose traveling salesman problem can be solved ``as efficiently" as the traveling salesman problem in $\mathbb{R}$. We also prove that if such metric spaces $X$ admit a biLipschitz embedding into $\mathbb{R}^n$, then $\mathsf{La}(X)$ admits a biLipschitz embedding into the product of $3n$ $\mathbb{R}$-trees.

Nonlinear type and metric embeddings of lamplighter spaces

Abstract

We prove that for all metric spaces the following properties of the lamplighter space are equivalent: (1) has finite Nagata dimension, (2) has Markov type 2, (3) does not contain the Hamming cubes with uniformly bounded biLipschitz distortion, (4) admits a weak biLipschitz embedding into a finite product of -trees. We characterize metric spaces for which satisfies properties (1)-(4) as those whose traveling salesman problem can be solved ``as efficiently" as the traveling salesman problem in . We also prove that if such metric spaces admit a biLipschitz embedding into , then admits a biLipschitz embedding into the product of -trees.

Paper Structure

This paper contains 14 sections, 21 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries
  4. Nagata dimension, $\mathbb{R}$-trees, and weak biLipschitz embeddings
  5. Doubling metric spaces
  6. Markov type
  7. Lamplighter Spaces
  8. Definition of $\mathop{\mathrm{TSP}}\nolimits$-efficient metric spaces
  9. Weak embeddings of lamplighters on $\mathop{\mathrm{TSP}}\nolimits$-efficient metric spaces
  10. Lamplighters over $\mathop{\mathrm{TSP}}\nolimits$-inefficient spaces
  11. The Main Result
  12. $d$-dominated line metrics on $(X,d)$ and biLipschitz embeddings of lamplighters on $X$
  13. Applications to lamplighters on grids
  14. Applications to lamplighters on TSP-efficient trees

Key Result

Theorem 1.1

(see Corollary cor:main) Let $X$ be a metric space and $\mathop{\mathrm{\mathsf{La}}}\nolimits(X)$ be the lampligther space on $X$. Then the following are equivalent. (i) $X$ is $\mathop{\mathrm{TSP}}\nolimits$-efficient (this is a new metric property that we describe below). (ii) $\mathop{\mathr

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 34 more