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}
}```
• Published 26 February 2016
• Mathematics
• arXiv: Metric Geometry
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…
6 Citations

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