Metric embeddings of Laakso graphs into Banach spaces

TL;DR

This paper constructs low-distortion metric embeddings of Laakso graphs into non-super-reflexive Banach spaces and $L_1[0,1]$, revealing geometric properties related to reflexivity and embedding distortions.

Contribution

It demonstrates that Laakso graphs can be embedded with arbitrarily small distortion into certain Banach spaces and provides lower bounds for embeddings into $L_1[0,1]$, advancing understanding of metric embedding theory.

Findings

Embeddings of Laakso graphs into non-super-reflexive Banach spaces with distortion less than 2+ε.

Embeddings into $L_1[0,1]$ with distortion 4/3.

Lower bounds of 9/8 and 5/4 for embeddings of $ ext{Laakso}_2$ and $D_2$ into $L_1[0,1]$.

Abstract

Let $X$ be Banach space which is not super-reflexive. Then, for each $n\ge1$ and $\varepsilon>0$, we exhibit metric embeddings of the Laakso graph $\mathcal{L}_n$ into $X$ with distortion less than $2+\varepsilon$ 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, $5/4$).