Distribution of the Volume of Weighted Gaussian Simplex

Abstract

Let $X_0,\dots ,X_l$ be independent standard Gaussian vectors in ${\mathbb{R}}^d$ such that $l⩽d$. We derive an explicit formula for the distribution of the volume of weighted Gaussian simplex without the origin -- $l$-dimensional simplex $\mathrm{conv}({\sigma }_0{X}_0,\dots ,{\sigma }_l{X}_l)$ (${\sigma }_0,\dots ,{\sigma }_l>0$).