Mixed random beta-polytopes

TL;DR

This paper extends the analysis of the average volume of random beta-polytopes to cases where vertices are independently beta-distributed with different parameters, providing new formulas and extending prior results.

Contribution

It generalizes existing results on beta-polytopes to non-identically distributed vertices and derives new expected volume formulas and facet distance functionals.

Findings

Derived an expression for the expected volume of generalized beta-polytopes.

Extended previous results to non-identically distributed beta vertices.

Computed the expected value of Wieacker's functional for these polytopes.

Abstract

In this paper, we generalize the result on the average volume of random polytopes with vertices following beta distributionsto the case of non-identically distributed vectors. Specifically,we consider the convex hull of independent random vectors in $\mathbb{R}^d$, where each vector follows a beta distribution with potentially different parameters. We derive an expression for the expected volume of these generalized beta--polytopes. Additionally, we compute the expected value of a functional introduced by Wieacker, which involves the distance of facets from the origin and their volumes.Our results extend the findings of Kabluchko, Temesvari,and Th\"ale. Key techniques used in the proofs include the Blaschke--Petkantschin formula, Kubota's formula, and projections of beta distributed random vectors.