Expected hyperbolic volumes of random beta polytopes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914520751931392 |
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| author | Kabluchko, Zakhar Schange, Philipp |
| author_facet | Kabluchko, Zakhar Schange, Philipp |
| contents | 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 $β_i \ge -1$: if $β_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\|x\|^2)^{β_i}$ on $\{\|x\|<1\}$, whereas the case $β_i=-1$ 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 $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) =
π\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Expected hyperbolic volumes of random beta polytopes Kabluchko, Zakhar Schange, Philipp Probability Metric Geometry Primary: 52A22, 60D05, Secondary: 26B15, 30B40, 52B11, 52A38, 52A55 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 $β_i \ge -1$: if $β_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\|x\|^2)^{β_i}$ on $\{\|x\|<1\}$, whereas the case $β_i=-1$ 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 $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = π\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \] |
| title | Expected hyperbolic volumes of random beta polytopes |
| topic | Probability Metric Geometry Primary: 52A22, 60D05, Secondary: 26B15, 30B40, 52B11, 52A38, 52A55 |
| url | https://arxiv.org/abs/2604.27793 |