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 every N>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 and all N⩾N(n,β), one has:(i)KN⊇c(β)Zq(K) for every q⩽ρ1ln(N/n), with probability greater than 1−c1exp(−c2N1−βnβ).(ii)For every q⩾ρ2ln(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 for all N⩽exp(n).