Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability
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arXiv
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| Format: | Preprint |
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2025
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| author | Kabluchko, Zakhar Steigenberger, David Albert |
| author_facet | Kabluchko, Zakhar Steigenberger, David Albert |
| contents | Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{β_i}1_{\{\|x\| <1\}}$. Here, $β_1,\ldots, β_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. We determine explicitly expected values of several functionals of these polytopes including the number of $k$-dimensional faces, the volume, the intrinsic volumes, the total $k$-volume of the $k$-skeleton, various angle sums, and the $S$-functional which generalizes and unifies many of the above examples. We identify and study the central object needed to analyze beta polytopes: beta cones. For these, we determine explicitly expected values of several functionals including the solid angle, conic intrinsic volumes and the number of $k$-dimensional faces. We identify expected conic intrinsic volumes of beta cones as a crucial quantity needed to express all the functionals mentioned above. We obtain a formula for these expected conic intrinsic volumes in terms of a function $Θ$ for which we provide an explicit integral representation. The proofs combine methods from integral and stochastic geometry with the study of the analytic properties of the function $Θ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability Kabluchko, Zakhar Steigenberger, David Albert Probability Metric Geometry Primary: 60D05, 52A22, Secondary: 52A55, 52B11, 52B05 Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{β_i}1_{\{\|x\| <1\}}$. Here, $β_1,\ldots, β_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. We determine explicitly expected values of several functionals of these polytopes including the number of $k$-dimensional faces, the volume, the intrinsic volumes, the total $k$-volume of the $k$-skeleton, various angle sums, and the $S$-functional which generalizes and unifies many of the above examples. We identify and study the central object needed to analyze beta polytopes: beta cones. For these, we determine explicitly expected values of several functionals including the solid angle, conic intrinsic volumes and the number of $k$-dimensional faces. We identify expected conic intrinsic volumes of beta cones as a crucial quantity needed to express all the functionals mentioned above. We obtain a formula for these expected conic intrinsic volumes in terms of a function $Θ$ for which we provide an explicit integral representation. The proofs combine methods from integral and stochastic geometry with the study of the analytic properties of the function $Θ$. |
| title | Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability |
| topic | Probability Metric Geometry Primary: 60D05, 52A22, Secondary: 52A55, 52B11, 52B05 |
| url | https://arxiv.org/abs/2503.22488 |