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.
Publié le : 2003-10-14
Classification:
Random polytopes,
Efron--Stein jackknife inequality,
approximation of convex bodies,
60D05,
52A22,
60C05,
60F15
@article{1068646381,
author = {Reitzner, Matthias},
title = {Random polytopes and the Efron--Stein jackknife inequality},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 2136-2166},
language = {en},
url = {http://dml.mathdoc.fr/item/1068646381}
}
Reitzner, Matthias. Random polytopes and the Efron--Stein jackknife inequality. Ann. Probab., Tome 31 (2003) no. 1, pp. 2136-2166. http://gdmltest.u-ga.fr/item/1068646381/