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Interaction of Poisson hyperplane processes and convex bodies

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arxiv 1812.08443 v1 pith:RHNJKRNO submitted 2018-12-20 math.PR

classification math.PR
keywords hyperplaneprocessconvexmeanpoissonpolytoperandomwidth
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abstract

Given a stationary and isotropic Poisson hyperplane process and a convex body $K$ in ${\mathbb R}^d$, we consider the random polytope defined by the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. We investigate how well the expected mean width of this random polytope approximates the mean width of $K$ if the intensity of the hyperplane process tends to infinity.

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  1. Poisson hyperplane processes and approximation of convex bodies

    math.PR 2019-08 accept novelty 7.0 of 10

    For Poisson hyperplane K-cells, the expected mean width excess is between n^{-1} log^{d-1} n and n^{-2/(d+1)}, with exact limits for smooth bodies and simplicial polytopes.

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