A Union of Euclidean Metric Spaces is Euclidean

@article{Makarychev2016AUO,
  title={A Union of Euclidean Metric Spaces is Euclidean},
  author={Konstantin Makarychev and Yury Makarychev},
  journal={arXiv: Metric Geometry},
  year={2016}
}
Suppose that a metric space $X$ is the union of two metric subspaces $A$ and $B$ that embed into Euclidean space with distortions $D_A$ and $D_B$, respectively. We prove that then $X$ embeds into Euclidean space with a bounded distortion (namely, with distortion at most $7D_A D_B + 2(D_A+D_B)$). Our result settles an open problem posed by Naor. Additionally, we present some corollaries and extensions of this result. In particular, we introduce and study a new concept of an "external bi… 

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