This is an announcement for the paper “Lipschitz Embeddings of Metric Spaces into $c_0$” by Florent P. Baudierhttps://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1, Robert Devillehttps://arxiv.org/find/math/1/au:+Deville_R/0/1/0/all/0/1.
Abstract: Let M be a separable metric space. We say that $f=(f_n): M\rightarrow c_0$ is a good-$\lambda$-embedding if, whenever $x, y\in M, x\neq y$ implies $d(x, y)\leq |f(x)-f(y)|$ and, for each $n, Lip(f_n)<\lambfda$, 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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.02025