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Inner Lipschitz approximation in o-minimal structures

Nhan Nguyen, Anna Valette, Guillaume Valette

Abstract

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.

Inner Lipschitz approximation in o-minimal structures

Abstract

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits cell decomposition, we show that the approximation can be required to be and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.
Paper Structure (8 sections, 12 theorems, 78 equations)

This paper contains 8 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. O-minimal structures.
  4. Stratifications
  5. Rugose vector fields
  6. Inner Lipschitz mappings
  7. Approximations and partitions of unity
  8. Approximation of Lipschitz definable mappings

Key Result

Lemma 3.2

Let $A\subset\mathbb{R}^n$ be a definable set and $f\in\mathcal{D}^0(A,\mathbb{R}^k)$. For all ${x_0}\in A$, we have

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 21 more