A discrete extension of the Blaschke Rolling Ball Theorem

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.