Random points and lattice points in convex bodies

We write K or Kd for the set of convex bodies in Rd, that is, compact convex sets with nonempty interior in Rd. Assume K ∈ K and x1, . . . , xn are random, independent points chosen according to the uniform distribution in K. The convex hull of these points, to be denoted by Kn, is called a random polytope inscribed in K. Thus Kn = [x1, . . . , xn] where [S] stands for the convex hull of the set S. The study of random polytopes began in 1864 with Sylvester’s famous “four-point question” [73]. Starting with the work of Rényi and Sulanke in 1964 [56] there has been a lot of research to understand the asymptotic behaviour of random polytopes. Most of it has been concentrated on the expectation of various functionals associated with Kn. For instance the number of vertices, f0(Kn), or more generally, the number of k-dimensional faces, fk(Kn), of Kn, or the volume missed by Kn, that is vol(K \Kn). The latter quantity measures how well Kn approximates K. As usual we will denote the expectation of fk(Kn) by Efk(Kn), and that of vol(K \Kn) by E(K,n). We write K1 for the set of those K ∈ K that have unit volume: volK = 1. This is convenient since then the Lebesgue measure and the uniform probability measure on K ∈ K1 coincide. The boundary of K is denoted by bdK. We write aff S for the affine hull of S. Assume a ∈ Rd is a unit vector and t ∈ R. Then the halfspace H = H(a ≤ t) is defined as H(a ≤ t) = {x ∈ R : a · x ≤ t}, where a · x is the scalar product of a and x. The bounding hyperplane of this halfspace is denoted by H(a = t). We will use often the Brunn-Minkowski theorem which says the following. If K,L ⊂ Rd are convex sets, then vol(K + L) ≥ (volK) + (volL) where K + L is the set of all k + l with k ∈ K and l ∈ L. For a proof see Schneider’s book [64] The Brunn-Minkowski theorem has an important consequence. Suppose K ∈ Kd, define h(t) = vol d−1K ∩H(a = t) and assume that h(t) is positive on an interval I.