ASYMPTOTIC APPROXIMATION OF SMOOTH CONVEX-BODIES BY POLYTOPES

We study asymptotic properties of the approximation of a sufficiently smooth convex body K in R(d) by the convex hulls of n points in the bo undary of K, for n --> infinity. The deviation is measured by the Haus dorff distance. The asymptotic distribution of the vertices of best-ap proximating polytopes is determined. Further results involve prescribe d densities for the vertices and describe the strength of approximatio n by either deterministic or random polytopes.