The monotonicity of f-vectors of random polytopes

Let K be a compact convex body in ℝ^d, let K_n be the convex hull of n points chosen uniformly and independently in K, and let fi(K_n) denote the number of i-dimensional faces of K_n. We show that for planar convex sets, is increasing in n. In dimension d≥3 we prove that if for some constants A and c>0 then the function is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.