# A discrete extension of the Blaschke Rolling Ball Theorem - Mathematics > Differential Geometry

A discrete extension of the Blaschke Rolling Ball Theorem - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: The Rolling Ball Theorem asserts that given a convex body K in Euclideanspace and having a smooth surface bdK with all principal curvatures notexceeding c>0 at all boundary points, K necessarily has the property that toeach boundary point there exists a ball B r of radius r=1-c, fully contained inK and touching bdK 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 anyboundary points x,y with |x-y|

Similarly, weconsider the dual type discrete Blaschke theorems ensuring certaincircumscribed polygons.

In the limit, the discrete theorem enables us toprovide a new proof for a strong result of Strantzen assuming only a.e.existence and lower estimations on the curvature.

Author: ** Sz.Gy.Re've'sz**

Source: https://arxiv.org/