Doubly random polytopes

Abstract

AbstractA 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 , 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.