On the characterization of some classes of proximally smooth sets

Abstract

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 ${\mathbb{R}}^n$.

19 pages, 3 figures

References in corpus (3)

• On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results
• A symmetry problem for the infinity Laplacian
• A new symmetry criterion based on the distance function and applications to PDE's

Cited by in corpus (9)

• On the structure of sets with positive reach
• On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results
• A symmetry problem for the infinity Laplacian
• Characterization of stadium-like domains via boundary value problems for the infinity Laplacian
• The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators
• Rigidity results for variational infinity ground states
• Delta-convex structure of the singular set of distance functions
• Principal eigenvalue problem for infinity Laplacian in metric spaces
• An occupation time formula for semimartingales in ${\mathbb{R}}^N$