Distribution of the Volume of Weighted Gaussian Simplex

TL;DR

This paper derives an explicit formula for the distribution of the volume of a weighted Gaussian simplex in high-dimensional space, extending understanding of Gaussian geometric structures.

Contribution

It provides a novel explicit formula for the volume distribution of weighted Gaussian simplices without the origin in -dimensional space.

Findings

Explicit formula for volume distribution of weighted Gaussian simplices

Applicable to high-dimensional Gaussian geometric analysis

Enhances understanding of Gaussian convex hulls

Abstract

Let $X_0, \ldots, X_l$ be independent standard Gaussian vectors in $\mathbb{R}^d$ such that $l \leqslant d$. We derive an explicit formula for the distribution of the volume of weighted Gaussian simplex without the origin -- $l$-dimensional simplex $\mathrm{conv}(\sigma_0X_0, \ldots, \sigma_lX_l)$ ($\sigma_0, \ldots, \sigma_l > 0$).