Random polytopes and the Efron--Stein jackknife inequality

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.