Mathematical Research Letters

Volume 18 (2011)

Number 6

A Characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability

Pages: 1179 – 1202



Jeehyeon Seo (Department of Mathematics, University of Illinois at Urbana Champaign, 1409 West Green Street, Urbana, IL, 61801, U.S.A.)


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 manifold 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 decomposition of an open subset $\Omega$ of the space. We prove that if the Whitney cubes embed uniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of $\Omega$ also embeds, then so does the full space.

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