A Union of Euclidean Metric Spaces is Euclidean

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 $X$ 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.