The monotonicity of f-vectors of random polytopes

TL;DR

This paper investigates the monotonicity of the expected number of faces of random convex hulls in various dimensions, showing increasing behavior in the plane and asymptotic increasing trends in higher dimensions.

Contribution

It establishes the monotonicity of the expected number of faces for random polytopes in the plane and provides asymptotic results for higher dimensions under certain conditions.

Findings

Expected number of vertices increases with n in 2D.

Number of facets asymptotically increases in higher dimensions.

Proof uses a random sampling argument.

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.