Asymptotics of the convex hull of spherical samples

TL;DR

This paper investigates the asymptotic properties of the convex hull formed by spherical samples in high-dimensional space, focusing on expected geometric features under specific distributional assumptions.

Contribution

It provides new asymptotic results for the expected number of vertices, facets, area, and volume of convex hulls for Gumbel max-domain distributions, with brief discussions on other models.

Findings

Asymptotic formulas for vertices and facets count

Expected area and volume asymptotics

Discussion on regularly varying distribution models

Abstract

In this paper we consider the convex hull of a spherically symmetric sample in $R^d$. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or $O$-regularly varying.