  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
$\mathbb{R}^n$.
