ASYMPTOTIC ESTIMATES FOR BEST AND STEPWISE APPROXIMATION OF CONVEX-BODIES .3.

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.