Poisson hyperplane processes and approximation of convex bodies

TL;DR

This paper investigates how well random polytopes generated by Poisson hyperplane processes approximate convex bodies, focusing on the convergence of certain geometric functionals as the process intensity increases.

Contribution

It analyzes the convergence behavior of the expected difference in functionals like hitting functional and mean width between the convex body and the random polytope as the hyperplane process intensity grows.

Findings

Convergence of the expected difference depends on the functional considered.

Order of convergence and limit relations are characterized for specific functionals.

Results provide insights into the approximation quality of random polytopes by Poisson hyperplanes.

Abstract

A natural model for the approximation of a convex body $K$ in $\mathbb{R}^d$ by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in the space, and consider the random polytope $Z_K$ defined as the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. If $f$ is a functional on convex bodies, then for increasing intensities of the process, the expectation of the difference $f(Z_K)-f(K)$ may or may not converge to zero. If it does, then the order of convergence and possible limit relations are of interest. We study these questions if $f$ is either the hitting functional or the mean width.