Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Maciej P. Denkowski

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).

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 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).

Paper Structure

This paper contains 4 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Lipschitz Implicit Function Theorem
  3. Local properties of Lipschitz functions satisfying condition $(Y)$.
  4. Acknowledgements

Key Result

Theorem 1.1

In the setting introduced above, assume that $k\leq n+1$ and for any choice of points $y_j$ in the convex hulls $\mathrm{cvx}(X_j)$, the system $y_1,\dots, y_k$ defines a $(k-1)$-dimensional simplex in $\mathbb{R}^n$. Then there is a neighbourhood $V$ of $a$ and a bi-Lipschitz homeomorphism $\phi\co

Theorems & Definitions (19)

  • Theorem 1.1: Y Theorem 1, Yomdin's Generic Structure Theorem
  • Theorem 1.2: Yomdin's LIFT
  • Theorem 2.1: Clarke, C2
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 9 more