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A discrete extension of the Blaschke rolling ball theorem

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Abstract

The rolling ball theorem asserts that given a convex body \(K\subset \mathbb {R}^d\) in Euclidean space and having a \(C^2\)-smooth surface \(\partial K\) with all principal curvatures not exceeding \(c>0\) at all boundary points, then 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 \(\partial K\) 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 \(\mathbf{x},\mathbf{y}\in \partial K\) with \(|\mathbf{x}-\mathbf{y}|\le \tau \), the angle \(\varphi (\mathbf{n}_\mathbf{x},\mathbf{n}_\mathbf{y}):= \arccos \langle \mathbf{n}_\mathbf{x},\mathbf{n}_\mathbf{y}\rangle \) of any unit outer normals \(\mathbf{n}_\mathbf{x},\mathbf{n}_\mathbf{y}\) at \(\mathbf{x}\) and at \(\mathbf{y}\), resp., does not exceed a given angle \(\varphi \). Then we construct a corresponding body, \(M(\tau ,\varphi )\), which is to lie fully within K while containing the given boundary point \(\mathbf{x}\in \partial K\). In dimension \(d=2\), that is, on the plane, 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. For \(d\ge 3\), directly we can derive only a weaker, quasi-precise form of the discrete inscribed ball theorem, while no space version of the circumscribed ball theorem is found. However, at least the higher dimensional smooth cases follow already from the plane versions of the smooth theorems, which obtain as limiting cases also from our discrete versions.

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References

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Acknowledgments

This paper was thoroughly checked by Endre Makai Jr., who gave numerous suggestions for improving the presentation. We thank to him also the sharpening of the inscribed mangled n-gon result, by means of the modulus of continuity with respect to arc length, as given above in Theorem 4.

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Correspondence to Szilárd Gy. Révész.

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Supported in part by the Hungarian National Foundation for Scientific Research, Project #s K-100461, NK-104183, K-109789 and K16-119638.

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Révész, S.G. A discrete extension of the Blaschke rolling ball theorem. Geom Dedicata 182, 51–72 (2016). https://doi.org/10.1007/s10711-015-0127-z

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