Random polytopes in convex bodies: Bridging the gap between extremal containers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912136780840960 |
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| author | Besau, Florian Gusakova, Anna Thäle, Christoph |
| author_facet | Besau, Florian Gusakova, Anna Thäle, Christoph |
| contents | We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of Euclidean balls of varying dimensions as a key example. As $n\to\infty$, we establish explicit growth rates for the expected number of facets, which depend in a subtle way on the the underlying model parameters. For the case of the uniform distribution, we further examine the expected number of faces of arbitrary dimensions as well as the volume difference. Our findings reveal that the family of random polytopes we introduce exhibits novel interpolative properties, bridging the gap between the classical extremal cases observed in the behavior of random polytopes within smooth versus polytopal convex containers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random polytopes in convex bodies: Bridging the gap between extremal containers Besau, Florian Gusakova, Anna Thäle, Christoph Probability Metric Geometry 52A22, 52A27, 60D05 We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of Euclidean balls of varying dimensions as a key example. As $n\to\infty$, we establish explicit growth rates for the expected number of facets, which depend in a subtle way on the the underlying model parameters. For the case of the uniform distribution, we further examine the expected number of faces of arbitrary dimensions as well as the volume difference. Our findings reveal that the family of random polytopes we introduce exhibits novel interpolative properties, bridging the gap between the classical extremal cases observed in the behavior of random polytopes within smooth versus polytopal convex containers. |
| title | Random polytopes in convex bodies: Bridging the gap between extremal containers |
| topic | Probability Metric Geometry 52A22, 52A27, 60D05 |
| url | https://arxiv.org/abs/2411.19163 |