A constructive proof of the Assouad embedding theorem with bounds on the dimension

TL;DR

This paper provides a constructive proof of the Assouad embedding theorem, demonstrating that doubling metric spaces can be bilipschitz embedded into Euclidean space with bounds on the dimension, depending only on the doubling constant.

Contribution

It offers a constructive proof of the Assouad embedding theorem with explicit bounds on the embedding dimension based on the metric's doubling constant.

Findings

Constructive proof of the Assouad embedding theorem.

Embedding dimension depends only on the doubling constant.

Valid for metric spaces with exponents in (1/2, 1).

Abstract

We give a constructive proof of a theorem of Naor and Neiman, (to appear, Revista Matematica Iberoamercana), which asserts that if $(E,d)$ is a doubling metric space, there is an integer $N > 0$, that depends only on the metric doubling constant, such that for each exponent $\alpha \in (1/2,1)$, we can find a bilipschitz mapping $F = (E,d^{\alpha}) \to \R^N$.