Bourgain's Theorem for a Union of Euclidean Metric Spaces

Abstract

We study the distortion of embedding a union of $k$ Euclidean sets into Euclidean space. We show that a union of $k$ Euclidean sets can be embedded with distortion at most $O(k^{1.128} \log k)$, independent of the number of points. This is an exponential improvement over the previous best known upper bound of $2^{O(k)}$ obtained by iterating the result of Makarychev and Makarychev (2016) for the special case $k = 2$. Surprisingly, with the additional assumption that each Euclidean set remains Euclidean when we adjoin any other two points, we can obtain an asymptotically tight distortion of $O(\log k)$, thus, providing a generalization of Bourgain's Theorem (1985). Along the way, we also show that the distortion of the union can be upper bounded by $O(\log k)$ times the maximum of the Lipschitz extension constants of individual Euclidean sets.