Metric embeddings of Laakso graphs into Banach spaces

Abstract

Let $X$ be Banach space which is not super-reflexive. Then, for each $n\ge 1$ and $\epsilon >0$, we exhibit metric embeddings of the Laakso graph ${\mathcal{L}}_n$ into $X$ with distortion less than $2+\epsilon$ and into $L_1[0,1]$ with distortion $4/3$. The distortion of an embedding of ${\mathcal{L}}_2$ (respectively, the diamond graph $D_2$) into $L_1[0,1]$ is at least $9/8$ (respectively, $5/4$).