On a Conjecture in Geometric Probability Regarding Asymptotic Normality of a Random Simplex

Abstract

A conjecture in geometric probability about the asymptotic normality of the $r$-content of the $r$-simplex, whose $r + 1$ vertices are independently uniformly distributed random points of which $p$ are in the interior and $r + 1 - p$ are on the boundary of a unit $n$-ball, is proved by Ruben (1977). In this article it is shown that the exact density of the random $r$-content is available in the most general case. The technique of inverse Mellin transform is used to get the exact density, thus requiring the knowledge of the $k$th moment of the $r$-content for all real $k$. This $k$th moment is already available in the literature. Approximations and asymptotic results as well as a simpler alternate proof for the conjecture are also given.