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1. Embedding Unions

If some collection of subsets of a Banach space embed into a Banach space with low distortion, is the same true of their union?


    1. Add remark

      Problem 1.1.

      [Bill Johnson] Suppose $A$ and $B$ are subsets of some metric space such that both $A$ and $B$ biLipschitzly embed into $L_1$. Does $A \cup B$ biLipschitzly embed into $L_1$?
        1. Remark. It was shown by Ostrovskii and Randrianantoanina that twisted unions of $L_1$-embeddable metric spaces also embed into $L_1$.


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              Problem 1.2.

              [Assaf Naor] Fix $D<\infty$. Suppose $A_1, A_2, \dots A_k$ are subsets of some metric space such that $c_2(A_i) \leq D$ for each $i$. What are the asymptotics (in terms of $k$) of $c_2(A_1 \cup A_2 \cup \dots A_k)$? Note that a trivial lower bound of $\Omega(\log k)$ is obtained by taking $A_i$ to be singleton sets in an expander.

                  Cite this as: AimPL: Metric embeddings, available at http://aimpl.org/metricembeddings.