A Union of Euclidean Metric Spaces is Euclidean

Konstantin Makarychev; Yury Makarychev

doi:10.19086/da.876

A Union of Euclidean Metric Spaces is Euclidean

Metric Geometry 2017-01-25 v3

Authors: 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.

Keywords

metric space Euclidean space distortion embedding external bi-Lipschitz extension metric geometry

Cite

@article{arxiv.1602.08426,
  title  = {A Union of Euclidean Metric Spaces is Euclidean},
  author = {Konstantin Makarychev and Yury Makarychev},
  journal= {arXiv preprint arXiv:1602.08426},
  year   = {2017}
}

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Reformatted for Discrete Analysis, updated metadata, and edited the title. This version is otherwise identical to the previous one

A Union of Euclidean Metric Spaces is Euclidean

Abstract

Keywords

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