Asymptotic shape of a random polytope in a convex body

Let K be an isotropic convex body in and let Z_q(K) be the L_q-centroid body of K. For every N>n consider the random polytope K_N:=conv{x₁,…,x_N} where x₁,…,x_N are independent random points, uniformly distributed in K. We prove that a random K_N 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 NN(n,β), one has:

(i) K_Nc(β)Z_q(K) for every qρ₁ln(N/n), with probability greater than 1−c₁exp(−c₂N^1−βn^β).

(ii) For every qρ₂ln(N/n), the expected mean width of K_N is bounded by c₃w(Z_q(K)).

As an application we show that the volume radius |K_N|^1/n of a random K_N satisfies the bounds for all Nexp(n).

Keywords: Convex body; Isotropic body; Isotropic constant; Random polytope; Centroid bodies; Mean width; Volume radius