Local quasi-isometries and tangent cones of definable germs

Abstract: In this paper, we introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a particular case of local quasi-isometric equivalence, we obtain Sampaio's tangent cone theorem as a corollary. As an application, we reprove the theorem by Fernandes-Sampaio, which states that the tangent cone of a Lipschitz normally embedded germ is also Lipschitz normally embedded.