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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.07952 |
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| _version_ | 1866912155689811968 |
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| author | Beck, Dominik |
| author_facet | Beck, Dominik |
| contents | New selected values of odd random simplex volumetric moments (moments of the volume of a random simplex picked from a given body) are derived in an exact form in various bodies in dimensions three, four, five, and six. In three dimensions, the well-known Efron's formula was used by Buchta & Reitzner and Zinani to deduce the mean volume of a random tetrahedron in a tetrahedron and a cube. However, for higher moments and/or in higher dimensions, the method fails. As it turned out, the same problem is also solvable using the Blashke-Petkantschin formula in Cartesian parametrisation in the form of the Canonical Section Integral (Base-height splitting). In our presentation, we show how to derive the older results mentioned above using our base-height splitting method and also touch on the essential steps of how the method translates to higher dimensions and for higher moments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Random Simplex Picking Beyond the Blashke Problem Beck, Dominik Metric Geometry Probability 60D05, 52A22 G.3 New selected values of odd random simplex volumetric moments (moments of the volume of a random simplex picked from a given body) are derived in an exact form in various bodies in dimensions three, four, five, and six. In three dimensions, the well-known Efron's formula was used by Buchta & Reitzner and Zinani to deduce the mean volume of a random tetrahedron in a tetrahedron and a cube. However, for higher moments and/or in higher dimensions, the method fails. As it turned out, the same problem is also solvable using the Blashke-Petkantschin formula in Cartesian parametrisation in the form of the Canonical Section Integral (Base-height splitting). In our presentation, we show how to derive the older results mentioned above using our base-height splitting method and also touch on the essential steps of how the method translates to higher dimensions and for higher moments. |
| title | On Random Simplex Picking Beyond the Blashke Problem |
| topic | Metric Geometry Probability 60D05, 52A22 G.3 |
| url | https://arxiv.org/abs/2412.07952 |