The Annals of Probability

Random polytopes and the Efron--Stein jackknife inequality

Matthias Reitzner

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Abstract

Let K be a smooth convex body. The convex hull of independent random points in K is a random polytope. Estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained. The essential step is the use of the Efron--Stein jackknife inequality for the variance of symmetric statistics. Consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope. A conjecture of Bárány concerning random and best-approximation of convex bodies is confirmed. Analogous results for random polytopes with vertices on the boundary of the convex body are given.

Article information

Source
Ann. Probab. Volume 31, Number 4 (2003), 2136-2166.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1068646381

Digital Object Identifier
doi:10.1214/aop/1068646381

Mathematical Reviews number (MathSciNet)
MR2016615

Zentralblatt MATH identifier
02077591

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]
Secondary: 60C05: Combinatorial probability 60F15: Strong theorems

Keywords
Random polytopes Efron--Stein jackknife inequality approximation of convex bodies

Citation

Reitzner, Matthias. Random polytopes and the Efron--Stein jackknife inequality. Ann. Probab. 31 (2003), no. 4, 2136--2166. doi:10.1214/aop/1068646381. http://projecteuclid.org/euclid.aop/1068646381.


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