On the variance of random polytopes (Q1959451)

This paper is devoted to so-called random polytopes. A random polytope is the convex hull of uniformly distributed random points in a convex body \(K\). A general lower bound on the variance of the volume and \(f\)-vector of random polytopes is proved. Also an upper bound in the case when \(K\) is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large numbers for the volume and \(f\)-vector of random polytopes when \(K\) is a polytope.