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On L 1 -Embeddability of Unions of L 1 -Embeddable Metric Spaces and of Twisted Unions of HypercubesAbstract 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 L p , 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in L p . This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to L p -spaces, 1 ≤ p < ∞.more » « less
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null (Ed.)Abstract Main results of the paper are as follows: (1) For any finite metric space $$M$$ the Lipschitz-free space on $$M$$ contains a large well-complemented subspace that is close to $$\ell _{1}^{n}$$ . (2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $$\ell _{1}^{n}$$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.more » « less
