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Differentiable approximation of continuous definable maps that preserves the image

Antonio Carbone

TL;DR

The work addresses whether a continuous definable map f: X -> Y between compact definable sets can be approximated by a definable map of class Cp without changing its image. It fuses o-minimal triangulations with PL geometry and introduces a surjective definable finite simplicial approximation, alongside an ε-squeezing mechanism, to control surjectivity. The main contributions are a surjective definable version of the finite simplicial approximation and a complete differentiable approximation theorem that preserves the image, extending approximation techniques in tame geometric settings. This yields image-preserving smoothing results for maps into singular or polyhedral targets, enriching the toolkit for faithful regularization in definable contexts.

Abstract

Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class $\mathcal{C}^p$ for each integer $p\geq 1$. In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image after the approximation. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a `surjective definable version' of the finite simplicial approximation theorem that we prove here.

Differentiable approximation of continuous definable maps that preserves the image

TL;DR

The work addresses whether a continuous definable map f: X -> Y between compact definable sets can be approximated by a definable map of class Cp without changing its image. It fuses o-minimal triangulations with PL geometry and introduces a surjective definable finite simplicial approximation, alongside an ε-squeezing mechanism, to control surjectivity. The main contributions are a surjective definable version of the finite simplicial approximation and a complete differentiable approximation theorem that preserves the image, extending approximation techniques in tame geometric settings. This yields image-preserving smoothing results for maps into singular or polyhedral targets, enriching the toolkit for faithful regularization in definable contexts.

Abstract

Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class for each integer . In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image after the approximation. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a `surjective definable version' of the finite simplicial approximation theorem that we prove here.
Paper Structure (9 sections, 10 theorems, 102 equations, 5 figures)

This paper contains 9 sections, 10 theorems, 102 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite simplicial complexes
  4. Pawłucki's desingularization
  5. Surjective simplicial approximation for definable maps.
  6. Surjective definable approximation between polyhedra
  7. $\epsilon$-squeezing map
  8. Proof of Proposition \ref{['key']}
  9. Differentiable approximation of continuous definable maps

Key Result

Theorem 1.2

Let $X$ be a compact definable subset of ${\mathbb R}^n$, $Y$ a definable subset of ${\mathbb R}^m$ and $f:X\to Y$ a continuous definable map. Let $\varepsilon>0$ and let $p\geq 1$ be an integer. Then, there exists a definable map $g:X\to Y$ of class ${\mathcal{C}}^p$ such that $\|f-g\|<\varepsilon$

Figures (5)

  • Figure 3.1: The polyhedron $|{\EuScript L}|\subset{\mathbb R}^2$.
  • Figure 3.2: The (open) star of a subcomplex ${\EuScript L}^*\subset{\EuScript L}$ (left) and the second (open) star of a vertex $\omega\in{\EuScript L}$ (right).
  • Figure 3.3: The values of a simplicial map $h$ on $\sigma_{\bullet}$ (left) and the corresponding values of $h^*$ on $\operatorname{sd}^2(\sigma)_{\bullet}$ (right). The red vertices are sent in $\omega_{i_1}$, the blue ones in $\omega_{i_2}$ and the green ones in $\omega_{i_3}$, where $\omega_{i_1}<_{{\EuScript L}}\omega_{i_2}<_{{\EuScript L}}\omega_{i_3}$.
  • Figure 3.4: The values of a simplicial map $h^*$ on $\operatorname{sd}^2(\sigma_k)_{\bullet}$ (left) and the corresponding values of $h$ (right). The red vertices are sent in $\omega_{0,k}$, the blue ones in $\omega_{1,k}$ and the green ones in $\omega_{2,k}$, where $\tau_k=\operatorname{Conv}(\{\omega_{0,k},\omega_{1,k},\omega_{2,k}\})$ and $\omega_{0,k}<_{{\EuScript L}}\omega_{1,k}<_{{\EuScript L}}\omega_{2,k}$.
  • Figure 5.1: A diagram that summarizes the current situation.

Theorems & Definitions (21)

  • Theorem 1.2: p1
  • Example 1.3
  • Theorem 1.4
  • Proposition 1.5: Surjective simplicial approximation
  • Theorem 2.1: Strict ${\mathcal{C}}^p$-refinement theorem, p1
  • Theorem 3.1: Finite simplicial approximation, mu
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.4
  • ...and 11 more