On the characterization of some classes of proximally smooth sets

We provide a complete characterization of closed sets with empty interior and positive reach in $\mathbb{R}^2$. As a consequence, we characterize open bounded domains in $\mathbb{R}^2$ whose high ridge and cut locus agree, and hence $C^1$ planar domains whose normal distance to the cut locus is constant along the boundary. The latter results extends to convex domains in $\mathbb{R}^n$.