The monotonicity of f-vectors of random polytopes

Abstract

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-1](Kn)) 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.