The convex hull of random points on the boundary of a simple polytope

TL;DR

This paper derives asymptotic formulas for the properties of convex hulls formed by random boundary points on simple polytopes, expanding understanding beyond interior point models.

Contribution

It provides the first rigorous asymptotic analysis of random polytopes on polytope boundaries, not limited to simple or simplicial cases.

Findings

Expected number of vertices and facets derived

Volume difference expectations calculated

Results contrast with interior point models

Abstract

The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are derived. This is one of the first investigations leading to rigorous results for random polytopes which are neither simple nor simplicial. The results contrast existing results when points are chosen in the interior of a convex set.