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October 2003 Random polytopes and the Efron--Stein jackknife inequality
Matthias Reitzner
Ann. Probab. 31(4): 2136-2166 (October 2003). DOI: 10.1214/aop/1068646381


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.


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Matthias Reitzner. "Random polytopes and the Efron--Stein jackknife inequality." Ann. Probab. 31 (4) 2136 - 2166, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1058.60010
MathSciNet: MR2016615
Digital Object Identifier: 10.1214/aop/1068646381

Primary: 52A22 , 60D05
Secondary: 60C05 , 60F15

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

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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