Affiliation:
1. Department of Mathematics and Computer Science , St. John’s University , Queens, NY , USA
2. Department of Mathematics , Miami University , Oxford , OH , USA
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 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 < ∞.
Subject
Applied Mathematics,Geometry and Topology,Analysis
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