Expected hyperbolic volumes of random beta polytopes

Abstract

Let $X_1,\dots ,X_n$ be independent random points in the closed unit ball of ${\mathbb{R}}^d$. Assume that each $X_i$ has a beta distribution with parameter ${\hat{I}}_i^2\ge -1$: if ${\hat{I}}_i^2>-1$, then $X_i$ has Lebesgue density proportional to $(1-\Vert x{\Vert }^2{)}^{{\hat{I}}_i^2}$ on $\{\Vert x\Vert <1\}$, whereas the case ${\hat{I}}_i^2=-1$ corresponds to the uniform distribution on the unit sphere $\{\Vert x\Vert =1\}$. Let $[X_1,\dots ,X_n]$ 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 $[X_1,\dots ,X_n]$. As a special case, if $X_1,\dots ,X_n$ are independent and uniformly distributed on the unit sphere in ${\mathbb{R}}^3$, then for every $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = π\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \]

26 pages