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

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, $1\le p<\infty$.