The monotonicity of $f$-vectors of random polytopes

Olivier Devillers,Guillaume Moroz,Matthias Reitzner,Marc Glisse,Xavier Goaoc

doi:10.1214/ecp.v18-2469

Abstract

Let $K$ be a compact convex body in ${\mathbb R}^d$, let $K_n$ be the convex hull of $n$ points chosen uniformly and independently in $K$, and let $f_{i}(K_n)$ denote the number of $i$-dimensional faces of $K_n$. ;We show that for planar convex sets, $E[f_0 (K_n)]$ is increasing in $n$. In dimension $d \geq 3$ we prove that if $\lim_{n \to \infty} \frac{E[f_{d-1}(K_n)]}{An^c}=1$ for some constants $A$ and $c>0$ then the function $n \mapsto E[f_{d-1}(K_n)]$ 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.

Highlights

What does a random polytope, that is, the convex hull of a finite set of random points in Rd, look like? This question goes back to Sylvester’s four point problem, which asked for the probability that four points chosen at random be in convex position
This paper present two contributions on the monotonicity of the f -vector of Kn
Our result is more general and applies to convex hulls of points i.i.d. from any “sufficiently generic” distribution and follows from a simple and elegant random sampling technique introduced by Clarkson [4] to analyze non-random geometric structures in discrete and computational geometry

Summary

Introduction

What does a random polytope, that is, the convex hull of a finite set of random points in Rd, look like? This question goes back to Sylvester’s four point problem, which asked for the probability that four points chosen at random be in convex position. It is known that Erf0pKnqs is always bounded from below by an increasing function of n, namely cpdq logd n where cpdq depends only on the dimension: this follows, via Efron’s formula [8], from a similar lower bound on the expected volume of Kn due to Bárány and Larman [1, Theorem 2]. While this is encouraging, it does not exclude the possibility of small oscillations preventing monotonicity. When general convex sets are considered, there may be more to this monotonicity question than meets the eye

Results

Monotonicity for convex domains in the plane

Random sampling