Lipschitz Embeddings of Metric Spaces into $c_0$

TL;DR

This paper characterizes when separable metric spaces can be embedded into c_0 with controlled Lipschitz properties, establishing a precise internal property that guarantees such embeddings, and extends previous results with simpler proofs.

Contribution

The paper introduces the internal property ppa(5) that characterizes the existence of good-5-embeddings into c_0 for separable metric spaces, extending prior work.

Findings

Existence of good-2-embeddings for all separable metric spaces.

Characterization of embeddings via the internal property ppa(5).

Simplified proofs of previous embedding results.

Abstract

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$\lambda$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<\lambda$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$\lambda$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $\pi(\lambda)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs.