Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Euclidean embedding, randomized clustering, and Lipschitz extension for finite and doubling subsets of $L_p$ when $p>2$

Assaf Naor, Kevin Ren

Abstract

Fix $p>2$. We prove that the Euclidean distortion of every $n$-point subset of $L_p$ is $p^3(\log n)^{\frac12+o(1)}$, thus, in particular, demonstrating that all $n$-point subsets of $L_p$ exhibit an asymptotic improvement over the $O(\log n)$ Euclidean distortion guarantee that Bourgain's embedding theorem provides for arbitrary $n$-point metric spaces. We also prove that the separation modulus of every $n$-point subset of $ L_p$ is $O(p^2\sqrt{\log n})$, which is sharp up to the dependence on $p$. We deduce from (a refinement of) this asymptotic evaluation of the finitary separation modulus of $ L_p$ that for any $n$-point subset $\mathcal{C}$ of $ L_p$, any Banach space $\mathbf{Z}$, and any $1$-Lipschitz function $f:\mathcal{C}\to \mathbf{Z}$, there exists a $O(p^2\sqrt{\log n})$-Lipschitz function $F:L_p\to \mathbf{Z}$ that extends $f$. We obtain analogous separation and extension statements for doubling subsets of $L_p$.

Euclidean embedding, randomized clustering, and Lipschitz extension for finite and doubling subsets of $L_p$ when $p>2$

Abstract

Fix . We prove that the Euclidean distortion of every -point subset of is , thus, in particular, demonstrating that all -point subsets of exhibit an asymptotic improvement over the Euclidean distortion guarantee that Bourgain's embedding theorem provides for arbitrary -point metric spaces. We also prove that the separation modulus of every -point subset of is , which is sharp up to the dependence on . We deduce from (a refinement of) this asymptotic evaluation of the finitary separation modulus of that for any -point subset of , any Banach space , and any -Lipschitz function , there exists a -Lipschitz function that extends . We obtain analogous separation and extension statements for doubling subsets of .

Paper Structure

This paper contains 22 sections, 24 theorems, 250 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bi-Lipschitz embeddings
  3. Toward a sharp Euclidean distortion growth rate for Lp
  4. Lipschitz extension
  5. Known Lipschitz extension impossibility results
  6. Lipschitz extension from doubling subsets
  7. Randomized clustering
  8. From separation of neighborhoods to Lipschitz extension
  9. From separation and extension to Euclidean embedding
  10. Localization and induction on scales, and a radial property of the Mazur map
  11. Beyond Lp
  12. Roadmap
  13. Measured descent for separated partitions (in the presence of Lipschitz extension)
  14. Weakly bi-Lipschitz embeddings
  15. From separation and extension to a weakly bi-Lipschitz embedding
  16. ...and 7 more sections

Key Result

Theorem 1.1

If $2<p<\infty$, then $L_p$ has nontrivial Euclidean distortion growth. More precisely,We will use throughout the ensuing text the following (standard) conventions for asymptotic notation, in addition to the usual $O(\cdot),o(\cdot),\Omega(\cdot), \Theta(\cdot)$ notation. Given $a,b>0$, by writing $

Figures (1)

  • Figure :

Theorems & Definitions (57)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Remark 1.11
  • Theorem 1.12
  • ...and 47 more