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Intrinsic volumes of inscribed random polytopes in smooth convex bodies
I. Bárány, F. Fodor, and V. Vígh
Source: Adv. in Appl. Probab. Volume 42, Number 3
(2010), 605-619.
Abstract
Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,...,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.
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Keywords: Convex body; intrinsic volume; uniform random polytope; variance; strong law of large numbers
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Permanent link to this document: http://projecteuclid.org/euclid.aap/1282924055
Digital Object Identifier: doi:10.1239/aap/1282924055
Mathematical Reviews number (MathSciNet): MR2779551
Zentralblatt MATH identifier: 1211.52005
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