On the Blaschke rolling disk theorem

TL;DR

This paper extends the classical Blaschke rolling disk theorem by providing curvature-based conditions for non-convex planar domains and introduces an algorithm to decompose any such domain into maximal rolling regions.

Contribution

It offers necessary and sufficient curvature conditions for rolling disks in non-convex domains, independent of convexity, and presents a decomposition algorithm.

Findings

Curvature-based criteria for non-convex domains.

Algorithm for domain decomposition into rolling regions.

Extension of classical convexity results.

Abstract

The Blaschke rolling disk theorem is a classical inclusion principle in differential geometry. This states that a planar convex domain whose boundary is a curve of class $C^2$ with (signed) curvature not exceeding a positive constant $\kappa$ is such that for each point on its boundary there exists a disk of radius ${1}/{\kappa}$ tangent to the boundary included in the closure of the domain. We describe geometric conditions relying exclusively on curvature and independent of any kind of convexity that allows us to give necessary and sufficient conditions for the existence of rolling disks for planar domains that are not necessarily convex. We finish by presenting an algorithm leading to a decomposition of any planar domain into a finite number of maximal rolling regions.