The Beauty of Random Polytopes Inscribed in the 2-sphere

TL;DR

This paper investigates the geometric properties of random polytopes inscribed in the 2-sphere, providing explicit formulas and elementary proofs for various metric characteristics, and extends results to ellipsoids.

Contribution

It offers new elementary proofs and explicit formulas for metric properties of random inscribed polytopes in the 2-sphere, including extensions to ellipsoids.

Findings

Explicit formulas for metric properties of random polytopes

Elementary proofs for properties in the 2-sphere case

Extension of results to ellipsoids with homeoid density

Abstract

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.