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arXiv · 1602.08426

A Union of Euclidean Metric Spaces is Euclidean

Konstantin Makarychev·Yury Makarychev
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Abstract

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-Lipschitz extension". In the end of the paper, we list a few related open problems.

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BibTeXRIS

Konstantin Makarychev, Yury Makarychev. 2017-01-23. A Union of Euclidean Metric Spaces is Euclidean. https://doi.org/10.19086/da.876

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