On L1-Embeddability of Unions of L1-Embeddable Metric Spaces and of Twisted Unions of Hypercubes

Abstract

Abstract We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L1-embeddable metric spaces also embed in L1 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 in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into Lp, 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in Lp. This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to Lp-spaces, 1 ≤ p < ∞.