A Union of Euclidean Metric Spaces is Euclidean
Konstantin Makarychev, Yury Makarychev

TL;DR
This paper proves that the union of two Euclidean-embeddable metric spaces also embeds into Euclidean space with bounded distortion, resolving an open problem and introducing new concepts like external bi-Lipschitz extension.
Contribution
It establishes a bound on Euclidean embedding distortion for unions of metric spaces and introduces the concept of external bi-Lipschitz extension.
Findings
Union of two Euclidean-embeddable spaces embeds with bounded distortion
Provides explicit distortion bounds based on subspace distortions
Introduces the concept of external bi-Lipschitz extension
Abstract
Suppose that a metric space is the union of two metric subspaces and that embed into Euclidean space with distortions and , respectively. We prove that then embeds into Euclidean space with a bounded distortion (namely, with distortion at most ). 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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