# Facets of spherical random polytopes

## TL;DR

This paper investigates the properties of facets of convex hulls formed by random points on high-dimensional spheres, focusing on facet height and count, and extends known fixed-dimension results to high-dimensional asymptotics.

## Contribution

It provides asymptotic formulas and regime classifications for the number and height of facets of spherical random polytopes as dimension grows, extending fixed-dimension results.

## Key findings

- Asymptotic formulas for the expected number of facets.
- Identification of regimes with different asymptotic behaviors.
- Extension of fixed-dimension results to high-dimensional settings.

## Abstract

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for $n$ and $d$ with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of some known results in fixed dimension to the case where dimension tends to infinity are described.