On the structure of sets with positive reach

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 $\varnothing \ne A\subset {\mathbb{R}}^d$ is a set of positive reach of topological dimension $0<k\le d$, then $A$ has its "$k$-dimensional regular part" $\varnothing \ne R\subset A$ 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.

Cited by in corpus (4)

• Structure of Submetries
• Metric thickenings of Euclidean submanifolds
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• A characterization of the local structure of two-dimensional sets with positive reach