Central limit theorems for random boundary polytopes
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915511368941568 |
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| author | Reitzner, Matthias Sonnleitner, Mathias |
| author_facet | Reitzner, Matthias Sonnleitner, Mathias |
| contents | The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if $k\ge 1$. We show that it is of order $n$. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order $n^{-1/2}$. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central limit theorems for random boundary polytopes Reitzner, Matthias Sonnleitner, Mathias Probability Metric Geometry 52A22 (Primary) 60D05 (Secondary) The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if $k\ge 1$. We show that it is of order $n$. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order $n^{-1/2}$. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions. |
| title | Central limit theorems for random boundary polytopes |
| topic | Probability Metric Geometry 52A22 (Primary) 60D05 (Secondary) |
| url | https://arxiv.org/abs/2509.20058 |