On the approximation of a polytope by its dual L_p-centroid bodies

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P ∈ Rⁿ by its dual L _p-centroid bodies is independent of the geometry of P. In particular, we show that if P has volume 1, lim _p→∞ p/log p (|Z _p^ο(P)|/|P^ο|-)= n². We provide an application to the approximation of polytopes by uniformly convex sets.