On the characterization of some classes of proximally smooth sets

TL;DR

This paper characterizes certain classes of proximally smooth sets in the plane, providing insights into their geometric properties and extending results to convex domains in higher dimensions.

Contribution

It offers a complete characterization of closed sets with empty interior and positive reach in 2, and relates these to properties of domains where the high ridge and cut locus coincide.

Findings

Characterization of closed sets with empty interior and positive reach in 2

Identification of conditions where high ridge and cut locus agree in planar domains

Extension of results to convex domains in n

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 $\mathbb{R}^n$.