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

TL;DR

This paper provides a complete local structural characterization of two-dimensional sets with positive reach in Euclidean space, extending previous results and offering new insights into their geometric properties.

Contribution

It offers a comprehensive characterization of 2D sets with positive reach and simplifies the proof of a related recent result, enhancing understanding of their local structure.

Findings

Characterization of local structure of 2D sets with positive reach

Sets with positive reach are locally contained in $C^{1,1}$ surfaces

2D sets with positive reach can be covered by countably many $C^{1,1}$ surfaces

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.