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.
Interaction of Poisson hyperplane processes and convex bodies
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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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2019 1verdicts
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Poisson hyperplane processes and approximation of convex bodies
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.