  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.
