A Characterization of Bi-Lipschitz Embeddable Metric Spaces in Terms of Local Bi-Lipschitz Embeddability

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-Carath\'eodory 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 {\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.