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

TL;DR

This paper proves that bi-Lipschitz homeomorphisms between subanalytic sets imply bi-Lipschitz equivalence of their tangent cones, with implications for the smoothness of Lipschitz regular complex analytic sets and invariance of directional dimensions.

Contribution

It establishes the bi-Lipschitz invariance of tangent cones for subanalytic sets and provides new insights into the structure of Lipschitz regular complex analytic sets.

Findings

Tangent cones are bi-Lipschitz homeomorphic under bi-Lipschitz maps.

Lipschitz regular complex analytic sets are smooth.

Alternative proof of invariance of directional dimensions.

Abstract

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.