RANDOM POLYTOPES IN SMOOTH CONVEX-BODIES

Let K subset-of R(d) be a convex body and choose points x1, x2,..., x(n) randomly, independently, and uniformly from K. Then K(n) = conv {x1,..., x(n)} is a random polytope that approximates K (as n --> infinity) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K - vol K(n) when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).