Bi-Lipschitz embeddings revisited

TL;DR

This paper investigates conditions under which metric spaces can be embedded into Euclidean space using bi-Lipschitz maps, focusing on the properties of the distance function and applications to compact metric-measure spaces.

Contribution

It provides three new sufficient conditions for bi-Lipschitz embeddings of compact metric-measure spaces into Euclidean space.

Findings

Identifies key properties of the distance function related to embeddings

Proposes three sufficient conditions for bi-Lipschitz embeddability

Enhances understanding of metric space embeddings into Euclidean spaces

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,\mu), we give three sufficient conditions for the existence of such a bi-Lipschitz embedding.