Random polytopes and the Efron-Stein jackknife inequality.
type: Item
instance of: scholarly article
title: Random polytopes and the Efron-Stein jackknife inequality.
zbMATH Open document ID: 1058.60010
author: Matthias Reitzner
published in: The Annals of Probability
publication date: 2004-07-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aop/1068646381
review text: For some \(d\geq2\), let \({\mathcal K}^2_+\) be the set of compact convex sets in \(d\)-dimensional Euclidean space with non-empty interior, positive Gaussian curvature and with boundary of differentiability class \({\mathcal C}^2\). For a fixed \(K\in{\mathcal K}^2_+\), choose points \(X_1,\ldots,X_n\) independently and uniformly at random. This paper is concerned with the number of vertices, \(N(K_n)\), and volume, \(V(K_n)\), of the convex hull \(K_n\) of the \(n\) points. Estimates are obtained for the variances of \(N(K_n)\) and \(V(K_n)\), from which strong laws of large numbers for these random variables follow. The essential step is the use of the jacknife inequality of \textit{B.~Efron} and \textit{C.~Stein} [Ann. Stat. 9, 586-596 (1981; Zbl 0481.62035)]. Results of \textit{I.~Bárány} [in: Second international conference in stochastic geometry, convex bodies and empirical measures. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 50, 43--50 (1997; Zbl 0890.52006)] concerning approximation of convex bodies by random polytopes and by best-approximating inscribed polytopes are strengthened. Analogous results for random polytopes with vertices on the boundary of the convex body are given.
reviewed by: M. P. Quine
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 2077591
zbMATH Keywords: random polytopes
zbMATH Keywords: Efron-Stein inequality
zbMATH Keywords: approximation of convex bodies
MaRDI profile type: Publication
OpenAlex ID: W1997708779
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DOI: 10.1214/AOP/1068646381
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