The Annals of Probability
- Ann. Probab.
- Volume 31, Number 4 (2003), 2136-2166.
Random polytopes and the Efron--Stein jackknife inequality
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.
Ann. Probab. Volume 31, Number 4 (2003), 2136-2166.
First available in Project Euclid: 12 November 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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.