Asymptotic shape of a random polytope in a convex body

Let K be an isotropic convex body in Rn and let Zq(K) be the Lq -centroid body of K. For everyN >n consider the random polytope KN := conv{x1, . . . , xN} where x1, . . . , xN are independent random points, uniformly distributed in K. We prove that a random KN is “asymptotically equivalent” to Z[ln(N/n)](K) in the following sense: there exist absolute constants ρ1,ρ2 > 0 such that, for all β ∈ (0, 12 ] and all N N(n,β), one has: (i) KN ⊇ c(β)Zq(K) for every q ρ1 ln(N/n), with probability greater than 1 − c1 exp(−c2N1−βnβ). (ii) For every q ρ2 ln(N/n), the expected mean width E[w(KN)] of KN is bounded by c3w(Zq(K)). As an application we show that the volume radius |KN|1/n of a random KN satisfies the bounds c4 √ln(2N/n) √n |KN|1/n c5LK √ln(2N/n) √n for all N exp(n). © 2009 Elsevier Inc. All rights reserved.