The minimum neighborliness of a random polytope

Abstract

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex hull of $S$? These polytopes are simplicial with probability one, but not much else is known except when more restrictive assumptions are imposed on $\mu$. In this paper we show that, with probability close to one, the convex hull of $S$ has a high degree of neighborliness no matter the underlying distribution $\mu$ as long as $n$ is not much bigger than $d$. As a concrete example, our result implies that if for each $d$ in $\mathbb{N}$ we choose a probability distribution $\mu_d$ on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and then set $P_n$ to be the convex hull of an i.i.d. sample of $n \le 5d/4$ random points from $\mu_d$, the probability that $P_n$ is $k$-neighborly approaches one as $d \to \infty$ for all $k\le d/20$. We also give a simple example of a family of distributions which essentially attain our lower bound on the $k$-neighborliness of a random polytope.