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 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 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 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 < ∞.