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

TL;DR

This paper proves Yomdin's Lipschitz Implicit Function Theorem using a geometric condition, clarifies its relation to Clarke's theorem, and enhances understanding of the medial axis in geometric analysis.

Contribution

It provides a rigorous proof of Yomdin's theorem, establishes its equivalence with Clarke's theorem, and extends the geometric understanding of medial axes.

Findings

Yomdin's Lipschitz Implicit Function Theorem is now proven and validated.

The geometric condition used is equivalent to Yomdin's original condition.

Additional properties of Lipschitz germs satisfying the Yomdin condition are established.

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).