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We characterize uniformly perfect, complete, doubling metric spaces whichembed bi- Lipschitzly into Euclidean space. Our result applies in particular tospaces of Grushin type equipped with Carnot-Carath\'eodory distance. Hence weobtain the first example of a sub-Riemannian mani- fold admitting such abi-Lipschitz embedding. Our techniques involve a passage from local to globalinformation, building on work of Christ and McShane. A new feature of our proofis the verification of the co-Lipschitz condition. This verification splitsinto a large scale case and a local case. These cases are distinguished by arelative distance map which is associated to a Whitney-type decom- position ofan open subset {\Omega} of the space. We prove that if the Whitney cubes embeduniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of{\Omega} also embeds, then so does the full space.

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