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

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 $\{g\le 1\}$ 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.