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<p>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 &amp; 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 Blaschke-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. </p>

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On Random Simplex Picking Beyond the Blaschke Problem

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Dominik Beck

摘要

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 Blaschke-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.