[Paper Review] A Characterization of Bi-Lipschitz Embeddable Metric Spaces in Terms of Local Bi-Lipschitz Embeddability

Jeehyeon Seo|arXiv (Cornell University)|May 12, 2011

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper provides a characterization of uniformly perfect, complete, doubling metric spaces that admit a bi-Lipschitz embedding into Euclidean space, showing that such embeddings exist if and only if the space supports a doubling measure and its complement relative to a closed subset admits uniformly local bi-Lipschitz embeddings. The key contribution is a global embedding result derived from local data via a Whitney-type decomposition and co-Lipschitz verification, with application to the Grushin plane, which is shown to admit such an embedding for the first time.

ABSTRACT

We characterize uniformly perfect, complete, doubling metric spaces which embed bi- Lipschitzly into Euclidean space. Our result applies in particular to spaces of Grushin type equipped with Carnot-Carathéodory distance. Hence we obtain the first example of a sub-Riemannian mani- fold admitting such a bi-Lipschitz embedding. Our techniques involve a passage from local to global information, building on work of Christ and McShane. A new feature of our proof is the verification of the co-Lipschitz condition. This verification splits into a large scale case and a local case. These cases are distinguished by a relative distance map which is associated to a Whitney-type decom- position of an open subset Ω of the space. We prove that if the Whitney cubes embed uniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of Ω also embeds, then so does the full space.

Motivation & Objective

• To characterize uniformly perfect, complete, doubling metric spaces that admit a bi-Lipschitz embedding into Euclidean space.
• To establish a sufficient condition for global bi-Lipschitz embeddability based on uniform local embeddings of Whitney-type cubes and a co-Lipschitz condition.
• To resolve the open question of whether the Grushin plane admits a bi-Lipschitz embedding into Euclidean space.
• To extend the framework of Christ and McShane by verifying the co-Lipschitz condition through a large-scale and local case distinction.

Proposed method

• Utilizes a Whitney-type decomposition of an open subset Ω in the metric space, derived from uniform perfectness and the existence of a doubling measure.
• Constructs local bi-Lipschitz embeddings on Whitney cubes Q** with uniform constants independent of the cube.
• Applies a coloring map to handle overlapping cubes and ensures uniform control over the global embedding via partition of unity.
• Verifies the co-Lipschitz condition by splitting the analysis into a large-scale case (based on relative distance map) and a local case.
• Combines the embeddings of the complement of Ω and the closed subset Y into Euclidean space to construct a global bi-Lipschitz embedding.
• Applies the framework to the Grushin plane and more general Grushin-type spaces with horizontal distributions defined by monomial vector fields.

Experimental results

Research questions

• RQ1 Under what conditions does a uniformly perfect, complete, doubling metric space admit a bi-Lipschitz embedding into Euclidean space?
• RQ2 Can the Grushin plane, equipped with Carnot-Carathéodory distance, be embedded bi-Lipschitzly into Euclidean space?
• RQ3 What is the role of the co-Lipschitz condition in passing from local to global bi-Lipschitz embeddings?
• RQ4 How can Christ-type local embeddings be uniformly controlled to yield a global embedding?
• RQ5 What are the sufficient geometric and analytic conditions on sub-Riemannian manifolds to ensure bi-Lipschitz embeddability?

Key findings

• A uniformly perfect, complete, doubling metric space admits a bi-Lipschitz embedding into Euclidean space if and only if it supports a doubling measure and the complement of a closed subset embeds uniformly locally with bi-Lipschitz constants independent of scale.
• The Grushin plane equipped with Carnot-Carathéodory distance admits a bi-Lipschitz embedding into some Euclidean space, providing the first example of a sub-Riemannian manifold with this property.
• The co-Lipschitz condition is verified by distinguishing a large-scale case and a local case using a relative distance map associated with the Whitney decomposition.
• Uniform local bi-Lipschitz embeddings on Whitney cubes Q** imply global bi-Lipschitz embeddability when combined with the embedding of the complement of Ω.
• The embedding dimension and bi-Lipschitz constant depend on the doubling constant of the measure, the embedding dimensions M₁ and M₂, and the local bi-Lipschitz constants.
• The result extends to more general Grushin-type spaces with horizontal distributions defined by monomials, suggesting a broader class of sub-Riemannian manifolds may admit such embeddings.