Asymptotic normality for random polytopes in non-Euclidean geometries

TL;DR

This paper proves asymptotic normality for the volume of random polytopes in non-Euclidean geometries, extending classical results to spherical, hyperbolic, and Hilbert geometries using Stein's method.

Contribution

It introduces new central limit theorems for random polytopes in non-Euclidean spaces, including Hilbert geometries, based on Stein's method and geometric analysis.

Findings

Asymptotic normality established for spherical and hyperbolic convex bodies.

Central limit theorems proved in Lutwak's dual Brunn--Minkowski theory.

Results derived from a CLT for weighted random polytopes in Euclidean spaces.

Abstract

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak's dual Brunn--Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein's method for normal approximation and geometric properties of weighted floating bodies.