On Yomdin's version of a Lipschitz Implicit Function Theorem and the geometry of medial axes

Abstract: In his beautiful paper on the central set from 1981, Y. Yomdin makes use of a Lipschitz Inverse Function Theorem that seemingly has been unproved until now. After a brief discussion of a natural and straightforward Lipschitz counterpart of an implicit function theorem, based on a geometric condition we finally provide a proof of Yomdin's version holds by proving the geometric condition is in fact equivalent to the one given by Yomdin. Therefore, Yomdin's Generic Structure Theorem, whose updated version is also presented here, concerning the medial axis (central set) of a subset of in ${\Rz}^n$ is now flawless. We also note that Yomdin's Lipschitz Implicit Function Theorem is equivalent to Clarke's Lipschitz Inverse Function Theorem. The paper ends with some additional properties of Lipschitz germs satisfying the Yomdin condition (e.g. a Lipschitz triviality result).