We consider approximations of a smooth convex body by inscribed and ci
rcumscribed convex polytopes as the number of vertices, resp. facets t
ends to infinity. The measure of deviation used is the difference of t
he mean width of the convex body and the approximating polytopes. The
following results are obtained. (i) An asymptotic formula for best app
roximation. (ii) Upper and lower estimates for step-by-step approximat
ion in terms of the so-called dispersion. (iii) For a sequence of best
approximating inscribed polytopes the sequence of vertex sets is unif
ormly distributed in the boundary of the convex body where the density
is specified explicitly.