On Random Simplex Picking Beyond the Blashke Problem

TL;DR

This paper derives exact formulas for higher moments of the volume of random simplices in various bodies across dimensions three to six, extending previous results and introducing a new base-height splitting method.

Contribution

It introduces a novel base-height splitting method to compute higher moments of random simplex volumes, extending known results to higher dimensions and moments.

Findings

Exact formulas for higher moments in dimensions 3-6

Extension of Blashke-Petkantschin approach using base-height splitting

Method applicability to higher dimensions and moments

Abstract

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.