  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.
