The monotonicity of $f$-vectors of random polytopes

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 $d≥3 we prove that if limn→∞ E[fd−1(Kn)]Anc=1$ for some constants A and c>0 then the function $n↦E[fd−1(Kn)]$ 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.