A discrete extension of the Blaschke Rolling Ball Theorem - Mathematics > Differential GeometryReport as inadecuate




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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|Then we construct acorresponding body, Mt,s, which is to lie fully within K while containing thegiven boundary point x.In dimension 2, M is almost a regular n-gon, and the result allows to recoverthe precise form of Blaschke-s Rolling Ball Theorem in the limit.
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/



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