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A characterization of the local structure of two-dimensional sets with positive reach

Jan Rataj, Ludek Zajicek

TL;DR

This work fully characterizes the local geometry of two-dimensional sets with positive reach in ${\mathbb R}^d$, extending earlier results and providing a constructive description near points where the tangent cone has dimension $1$ or $2$. The authors develop an elementary, Whitney-extension–based approach to Lipschitz the tangent-dimension map $\psi_k^A$ and demonstrate that tangent strata lie on ${\it C}^{1,1}$ surfaces, yielding a countable cover of ${\mathbb R}^d$-embedded sets by such surfaces. A central contribution is the germ-based framework for two-dimensional sets, employing ${\bf B}$-sets and multirotations to model local neighborhoods, along with a novel use of product-integrals to control deformations. The results lead to corollaries that compact two-dimensional sets with positive reach admit locally bi-Lipschitz parametrizations and that, in general, 2D positive-reach sets can be reconstructed from well-structured local pieces, with potential implications for data analysis and manifold reconstruction. An alternative, elementary proof of Lytchak's theorem L is also provided, strengthening the accessibility of the local-structure description in higher dimensions.

Abstract

The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in $R^d$. We also present a more elementary proof of a recent result of A. Lytchak which describes for $k\leq d$ the local structure of $k$-dimensional sets with positive reach $A$ in $R^d$ at points where the tangent cone of $A$ is $k$-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in $R^d$. Our method also shows that, for any set $A\subset R^d$ with positive reach, the set of points at which the tangent cone of $A$ is $k$-dimensional is locally contained in a $k$-dimensional $C^{1,1}$ surface. As a consequence we obtain that if $1\leq k<d$, and $A$ is $k$-dimensional, it can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces.

A characterization of the local structure of two-dimensional sets with positive reach

TL;DR

This work fully characterizes the local geometry of two-dimensional sets with positive reach in , extending earlier results and providing a constructive description near points where the tangent cone has dimension or . The authors develop an elementary, Whitney-extension–based approach to Lipschitz the tangent-dimension map and demonstrate that tangent strata lie on surfaces, yielding a countable cover of -embedded sets by such surfaces. A central contribution is the germ-based framework for two-dimensional sets, employing -sets and multirotations to model local neighborhoods, along with a novel use of product-integrals to control deformations. The results lead to corollaries that compact two-dimensional sets with positive reach admit locally bi-Lipschitz parametrizations and that, in general, 2D positive-reach sets can be reconstructed from well-structured local pieces, with potential implications for data analysis and manifold reconstruction. An alternative, elementary proof of Lytchak's theorem L is also provided, strengthening the accessibility of the local-structure description in higher dimensions.

Abstract

The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in . We also present a more elementary proof of a recent result of A. Lytchak which describes for the local structure of -dimensional sets with positive reach in at points where the tangent cone of is -dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in . Our method also shows that, for any set with positive reach, the set of points at which the tangent cone of is -dimensional is locally contained in a -dimensional surface. As a consequence we obtain that if , and is -dimensional, it can be covered by countably many -dimensional surfaces.

Paper Structure

This paper contains 15 sections, 39 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions
  4. Functions, mappings and surfaces
  5. Basic and auxiliary results on sets of positive reach
  6. Lipschitzness of $\psi^A_k$ on some subsets of $T_k(A)$
  7. Local structure of $T_k(A)$
  8. Local structure of a $2$-dimensional set $A$ at points of $T_1(A)$
  9. Description of the strategy and some auxiliary notions
  10. Proof of Proposition \ref{['T1']}
  11. Some properties of skewered sets with positive reach
  12. Product-integrals of operators and the proof of Proposition \ref{['T2']}
  13. Proof of Proposition \ref{['T3']}
  14. Proofs of Theorem \ref{['T_main']} and Corollary \ref{['bilip']}
  15. An alternative proof of Theorem L

Key Result

Theorem 1.3

Let $A\subset\mathbb{R}^d$ ($d\geq 3$) be a set with positive reach with $\dim A\leq 2$ and let $a\in A$ be such that $\dim(\operatorname{Tan}(A,a))=1$. Then there exist a $\operatorname{B}$-set $B\subset\mathbb{R}^2\subset\mathbb{R}^d$, a multirotation $\rho:\mathbb{R}^d\to\mathbb{R}^d$ associated

Theorems & Definitions (94)

  • Definition 1.1: B-sets
  • Definition 1.2: multirotations
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 2.1
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 84 more