Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.