Beta polytopes and Poisson polyhedra: $f$-vectors and angles

TL;DR

This paper investigates the geometric properties of random polytopes generated from beta and beta' distributions, providing exact and asymptotic formulas for face counts, angles, and intrinsic volumes, with applications to Poisson hyperplane tessellations.

Contribution

It introduces new explicit formulas for face numbers and angles of beta and beta' polytopes, and connects these to Poisson hyperplane tessellations and extreme-value theory.

Findings

Expected number of faces increases with sample size n.

Derived formulas for internal and external angles of the polytopes.

Established limit theorems for high-dimensional behavior.

Abstract

We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,\beta} (x) = c_{d,\beta} (1-\|x\|^2)^{\beta}, \qquad \|x\| < 1, \quad \text{(beta distribution, $\beta>-1$)} $$ or $$ \tilde f_{d,\beta} (x) = \tilde{c}_{d,\beta} (1+\|x\|^2)^{-\beta}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution, $\beta>d/2$)}. $$ This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of $k$-faces of $[X_1,\ldots,X_n]$ for arbitrary $k\in\{0,1,\ldots,d-1\}$. We prove that for any such $k$ this expected number is strictly monotonically increasing with $n$. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large $n$ limit in the beta' case, we compute the expected $f$-vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of $k$-faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in $\mathbb R^d$. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson--Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this $f$-vector in the high-dimensional regime, as $d\to\infty$. Finally, we relate the $d$-dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory.