Expected extremal area of facets of random polytopes

TL;DR

This paper investigates the expected maximum and minimum surface areas of facets of spherical random polytopes, providing asymptotic growth rates in fixed dimensions, which enhances understanding of their extremal geometric properties.

Contribution

It determines the asymptotic behavior of extremal facet surface areas of spherical random polytopes in fixed dimensions, a novel analysis in stochastic convex geometry.

Findings

Asymptotic growth rates of extremal facet surface areas are established.

Results apply to convex hulls of points on the unit sphere in fixed dimensions.

Provides bounds up to absolute constants for these extremal properties.

Abstract

We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in $\mathbb{R}^n$. The extremal properties of interest are the expected values of the maximum and minimum surface area among facets. We determine the asymptotic growth in every fixed dimension, up to absolute constants.