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

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-Carathodory 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 of the space. We prove that if the Whitney cubes embed uniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of also embeds, then so does the full space.