EXPECTATION OF RANDOM POLYTOPES

A random polytope P-n in a convex body C is the convex hull of n ident ically and independently distributed points in C. Its expectation is a convex body in the interior of C. We study the deviation of the expec tation of P-n from C as n --> infinity : while for C of class C-k+1,k greater than or equal to 1, precise asymptotic expansions for the devi ation exist, the behaviour of the deviation is extremely irregular for most convex bodies C of class C-1.