Doubly random polytopes

TL;DR

This paper introduces a two-step probabilistic model for generating random polytopes, analyzing their approximation of the sphere and their combinatorial complexity based on parameters like hyperplanes, sampling probability, and dimension.

Contribution

It provides new theoretical results on the approximation quality and asymptotic complexity of the generated polytopes within this random model.

Findings

Q approximates the unit sphere as m and p vary

Asymptotic behavior of the number of vertices and facets

Conditions under which the polytope complexity grows

Abstract

A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in $\mathbb{R}^d$, and the second step is to sample each vertex of $P$ independently with probability $p$ and let $Q$ be the convex hull of the sampled vertices. We establish results on how well $Q$ approximates the unit sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial complexity of $Q$ for certain regimes of $p$.