We derive an asymptotic formula for the difference of the volumes of a
smooth convex body and its inscribed polytopes of maximum volume as t
he number of vertices tends to infinity. A similar result is proved fo
r circumscribed polytopes. In addition, step-by-step approximation res
ults are presented. The proofs are based on Delone triangulations, res
p. Dirichlet - Voronoi tilings and make use of a generalized version o
f Blaschke's ''Schuttelung''; they are quite involved.