Approximation of convex bodies and a momentum lemma for power diagrams

Let C be a convex body in Euclidean d-space IE , i.e., a compact convex set with non-empty interior, and denote by P i n and P (n) the set of polytopes with at most n vertices inscribed to C and the set of polytopes with at most n facets circumscribed to C, respectively. Denote by δ(., .) the symmetric difference metric. Beginning with the work of L. Fejes Tóth [2], there are many investigations (cf. the survey [5]) on the asymptotic behavior as n→∞ of the distance of C to its best approximating polytopes with at most n vertices or facets, i.e., of