Intrinsic volumes and Gaussian polytopes: the missing piece of the jigsaw

The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this missing piece of the jigsaw in the theory of Gaussian polytopes are discussed

Random points and lattice points in convex bodies

Assume K ⊂ Rd is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester’s famous four-point problem and from the theory of random polytopes. The second case is when X = K ∩ Zd where Zd is the lattice of integer points in Rd and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar