The minimum neighborliness of a random polytope

TL;DR

This paper demonstrates that random polytopes formed from points sampled from broad distributions exhibit high neighborliness with high probability when the number of points is not much larger than the dimension, revealing fundamental geometric properties.

Contribution

It establishes that random convex hulls are highly neighborly under minimal assumptions on the distribution, extending understanding beyond restrictive cases.

Findings

High neighborliness occurs with high probability for random polytopes when n is close to d.

The probability that a random polytope is k-neighborly approaches one as dimension increases.

A family of distributions is provided that nearly attains the lower bound on neighborliness.

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.