A discrete extension of the Blaschke Rolling Ball Theorem

TL;DR

This paper develops a discrete analogue of the Blaschke Rolling Ball Theorem in the plane, relating boundary curvature conditions to inscribed and circumscribed polygons, and offers a new proof of a curvature-related theorem.

Contribution

It introduces a discrete curvature condition and constructs bodies that approximate the smooth case, extending the classical theorem to a discrete setting and providing new proofs for existing results.

Findings

Discrete bodies approximate smooth convex bodies as parameters tend to zero.

The discrete theorem recovers the classical Blaschke Rolling Ball Theorem in the limit.

New proof of Strantzen's curvature result using discrete geometric methods.

Abstract

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary point there exists a ball B_r of radius r=1/c, fully contained in K and touching bd(K) at the given boundary point from the inside of K. In the present work we prove a discrete analogue of the result on the plane. We consider a certain discrete condition on the curvature, namely that to any boundary points x,y with |x-y|<t, the angle between any unit outer normals at x and at y, resp., does not exceed a given angle s. Then we construct a corresponding body, M(t,s), which is to lie fully within K while containing the given boundary point x. In dimension 2, M is almost a regular n-gon, and the result allows to recover the precise form of Blaschke's Rolling Ball Theorem in the limit. Similarly, we consider the dual type discrete Blaschke theorems ensuring certain circumscribed polygons. In the limit, the discrete theorem enables us to provide a new proof for a strong result of Strantzen assuming only a.e. existence and lower estimations on the curvature.