On $L_1$-embeddability of unions of $L_1$-embeddable metric spaces and of twisted unions of hypercubes

TL;DR

This paper investigates how certain unions of metric spaces, especially those embeddable in L_1, can themselves be embedded in L_1 with bounded distortion, and provides new examples of metric spaces with high embedding distortion.

Contribution

It proves that twisted unions of L_1-embeddable spaces embed in L_1 with bounded distortion under mild conditions, answering a question by Naor and Rabani, and extends high-distortion embedding results to L_p spaces.

Findings

Twisted unions of L_1-embeddable spaces embed in L_1 with bounded distortion.

Constructs new metric space examples with at least 3 distortion for embeddings into L_p.

Extends high-distortion embedding results from Hilbert spaces to L_p spaces.

Abstract

We study properties of twisted unions of metric spaces introduced by Johnson, Lindenstrauss, and Schechtman, and by Naor and Rabani. In particular, we prove that under certain natural mild assumptions twisted unions of $L_1$-embeddable metric spaces also embed in $L_1$ with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated by Naor and by Naor and Rabani. In the second part of the paper we give new simple examples of metric spaces such their every embedding into $L_p$, $1\le p<\infty$, has distortion at least $3$, but which are a union of two subsets, each isometrically embeddable in $L_p$. This extends an analogous result of K.~Makarychev and Y.~Makarychev from Hilbert spaces to $L_p$-spaces, $1\le p<\infty$.