Non-negative forms, volumes of sublevel sets, complete monotonicity and moment matrices

TL;DR

This paper explores properties of positive degree forms, showing the volume of their sublevel sets is completely monotone, characterizing forms with finite volume, and linking Gaussian moments to covariance matrices.

Contribution

It introduces the complete monotonicity of sublevel set volumes, characterizes forms with finite volume, and connects Gaussian moments to covariance matrices.

Findings

Volume of sublevel sets is completely monotone on the cone.

Characterization of forms with finite Lebesgue volume.

Connection between Gaussian moments and inverse covariance matrix.

Abstract

Let $\mathcal{C}_{d,n}$ be the convex cone consisting of real $n$-variate degree $d$ forms that are strictly positive on $\mathbb{R}^n\setminus \{\mathbf{0}\}$. We prove that the Lebesgue volume of the sublevel set $\{g\leq 1\}$ of $g\in \mathcal{C}_{d,n}$ is a completely monotone function on $\mathcal{C}_{d,n}$ and investigate the related properties. Furthermore, we provide (partial) characterization of forms, whose sublevel sets have finite Lebesgue volume. Finally, we discover an interesting property of a centered Gaussian distribution, establishing a connection between the matrix of its degree $d$ moments and the quadratic form given by the inverse of its covariance matrix.