On the structure of sets with positive reach

TL;DR

This paper characterizes the structure of sets with positive reach in Euclidean spaces, revealing their decomposition into regular manifold parts and lower-dimensional surfaces, with implications for geometric measure theory.

Contribution

It provides a complete geometric characterization of sets with positive reach in the plane and higher dimensions, including their decomposition into smooth and DC surfaces.

Findings

Sets with positive reach have a regular $C^{1,1}$ manifold part.

The remaining set can be covered by finitely many DC surfaces.

Boundaries of such sets can be locally covered by semiconcave hypersurfaces.

Abstract

We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset \neq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces.