Minkowski weak embedding theorem

TL;DR

This paper extends the classical Assouad embedding theorem to non-injective weak embeddings for spaces with finite Minkowski dimension, enabling the study of non-doubling spaces like those in random geometry within Euclidean spaces.

Contribution

It introduces a non-injective analog of the Assouad theorem for Minkowski dimension spaces, broadening the scope of embeddings to non-doubling metric spaces.

Findings

Established a weak embedding theorem for Minkowski dimension spaces.

Applied the result to spaces in random geometry and mathematical physics.

Enabled Euclidean analysis of non-doubling metric spaces.

Abstract

A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem for spaces of finite Minkowski dimension. This allows for non-doubling spaces to be weakly embedded and studied in the usual Euclidean setting. Such spaces often arise in the context of random geometry and mathematical physics with the Brownian continuum tree and Liouville quantum gravity metrics being prominent examples.