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Approximation of smooth convex bodies by random circumscribed polytopes. (English) Zbl 1049.60009

Authors’ summary: Choose \(n\) independent random points on the boundary of a convex body \(K\subset\mathbb{R}^d\). The intersection of the supporting halfspaces of these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean width are investigated. On the case that the boundary of \(K\) is sufficiently smooth, asymptotic expansions as \(n\to\infty\) are derived even in the case when the curvature is allowed to be zero. We compare our results to the analogous results for best approximating polytopes.

MSC:

60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
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