Asymptotic approximation of smooth convex bodies by general polytopes

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we want to derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed. Let Pn and P(n) denote the set of polytopes with at most n vertices and n facets, respectively. For a convex body C, i.e., a compact convex set with non-empty interior, we are interested in the asymptotic behavior as n→∞ of