Random inscribed polytopes in projective geometries

TL;DR

This paper proves central limit theorems for the volumes and widths of random inscribed polytopes in projective geometries, extending classical results to more general geometric settings.

Contribution

It introduces a general CLT for weighted volumes of convex hulls of random boundary points in Euclidean space, applicable to projective geometries.

Findings

Central limit theorems for volumes of random inscribed polytopes in projective geometries.

Normal approximation results for dual volumes and mean width of random polyhedral sets.

Geometric estimates and Berry-Esseen bounds underpin the theoretical results.

Abstract

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and Berry-Esseen bounds for functionals of independent random variables.