A union of Euclidean metric spaces is Euclidean

Konstantin Makarychev, Yury Makarychev

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Suppose that a metric space X is the union of two metric subspaces A and B that embed into Euclidean space with distortions DA and DB, respectively. We prove that then X embeds into Euclidean space with a bounded distortion (namely, with distortion at most 7DADB+2(DA+DB)). 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.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalDiscrete Analysis
Issue number2016
StatePublished - 2016


  • Lipschitz extension
  • Local-global properties
  • Metric geometry

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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