Expected hyperbolic volumes of random beta polytopes

TL;DR

This paper derives explicit formulas for the expected hyperbolic volume of random beta polytopes formed by points in a unit ball, connecting probability distributions with hyperbolic geometry.

Contribution

It provides the first closed-form formulas for the expected hyperbolic volume of random polytopes with beta-distributed vertices in the Klein model.

Findings

Explicit formulas for expected hyperbolic volume of beta polytopes.

Special case formula for uniform sphere distribution in 3D.

Connection between beta distributions and hyperbolic geometric volumes.

Abstract

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $\beta_i \ge -1$: if $\beta_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\|x\|^2)^{\beta_i}$ on $\{\|x\|<1\}$, whereas the case $\beta_i=-1$ corresponds to the uniform distribution on the unit sphere $\{\|x\|=1\}$. Let $[X_1,\ldots,X_n]$ denote the convex hull of these points. Interpreting the unit ball as the Klein model of hyperbolic geometry, we derive closed-form formulas for the expected hyperbolic volume of the random hyperbolic polytope $[X_1,\ldots,X_n]$. As a special case, if $X_1,\ldots,X_n$ are independent and uniformly distributed on the unit sphere in $\mathbb{R}^3$, then for every $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = \pi\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \]