Expected hyperbolic volumes of random beta polytopes
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 $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = \pi\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \]