Polytopal approximation bounding the number of k-faces

Assume that M is a convex body with C-2 boundary in R-d. The paper consider s polytopal approximation of M with respect to the most commonly used metri cs, like the symmetric difference metric delta(S), the L-p metric, 1 less t han or equal to p less than or equal to infinity, or the Banach-Mazur metri c. In case of delta(S), the main result states that if P-n is a polytope wh ose number of k faces is at most n then delta(S)(M, P-n)>1/67e(2)pi.1/d.(integral(partial derivative M) kappa(x)(1/ (d+1)) dx)((d+1)/(d-1)).1/n(2/(d-1).) The analogous estimates are proved for all the other metrics. Finally, the optimality of these estimates is verified up to a constant depending on the metric and the dimension. (C) 2000 Academic Press.