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.