On the variance of 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. 1 The main results Let K ⊂ IR be a convex set of volume one. Assume x1, . . . , xn is a random sample of n independent, uniform points from K. The random polytope Kn is just the convex hull of these points: Kn = [x1, . . . , xn]. It is one of the classical problems in stochastic geometry to investigate the asymptotic behaviour of Kn, see, e.g., the book of Kendall and Moran [14], and the recent book on stochastic geometry of Schneider and Weil [20]. Starting with Rényi and Sulanke [16] in 1963, there have been many results concerning the expectation of various functionals of Kn. For instance, the expectation of the volume V (Kn), and of the number, f`(Kn), of `-dimensional faces of Kn (` = 0, . . . , d − 1) have been determined, see [23] for an extensive survey, and also [7] for more recent results. Yet determining the variance is in general still an open problem. For smooth convex bodies this has been solved, up to order of magnitude, by Reitzner [17] and [19], extending an earlier upper bound, for the case of the unit ball, by Küfer [15] (and some other sporadic results in dimension 2). Recently Schreiber and Yukich [21] have determined the precise asymptotic behaviour of the variance of f0(Kn) when K is the unit ball, a significant breakthrough. On the other hand for convex polytopes much less is known, and it seems that the situation there is much more delicate. In this case we denote the underlying polytope by P instead of K and the random polytope by Pn. In the planar case, variances and central limit theorems for f0(Pn) and V (Pn) were proved by Groeneboom [12], and Cabo and Groeneboom [9], but it seems that the stated variances are incorrect (see the discussion in Buchta [8]). In this paper we determine the order of magnitude of the variance of the volume and the number of `-dimensional faces of the random polytope when the mother body