Polytopal Approximation Bounding the Number of k-Faces Original Research Article

Assume that M is a convex body with C2 boundary in Rd. The paper considers polytopal approximation of M with respect to the most commonly used metrics, like the symmetric difference metric δS, the Lp metric, 1⩽p⩽∞, or the Banach–Mazur metric. In case of δS, the main result states that if Pn is a polytope whose number of k faces is at most n then δS(M, Pn)>167e2π·1d·∫∂M κ(x)1/(d+1) dx(d+1)/(d−1) ·1n2/(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.