Let \(\mathcal{C}_{d,n}\) be the convex cone consisting of \(n\)-variate degree \(d\) homogeneous polynomials with real coefficients that are strictly positive on \(\mathbb{R}^n \setminus \{ \mathbf{0} \}\).
In Theorem 1.1, the authors not only find the representation of the volume function \(f: \mathcal{C}_{d,n} \rightarrow \mathbb{R}\) defined as \(f(g) = \text{vol}(\{ g \leq 1 \})\) for \(g \in \mathcal{C}_{d,n}\), but they also prove that the Lebesgue volume of the sublevel set \(\{ g \leq 1 \}\) defines a completely monotone function on \(\mathcal{C}_{d,n}\). In addition, they demonstrate that for a nonnegative form \(h \in \overline{\mathcal{C}_{e,n}}\) of degree \(e\geq 0\), the function \(f_h: \mathcal{C}_{d,n} \rightarrow \mathbb{R}\) given by \(f_h(g) = \int_{\{ g \leq 1 \}} h(\mathbf{x}) d\mathbf{x}\) is monotone that generalize Theorem 1.1. The authors also discuss related properties and prove a version of Theorem 1.1 for sum of squares forms.
Furthermore, they investigate the cone of nonnegative forms with sublevel sets of finite volumes. Let \(\mathcal{V}_{d,n}\) be the set of \(n\)-variate homogeneous polynomials \(g\) of degree \(d\) such that the volume \(\text{vol}(\{ g \leq 1 \})\) is finite. Note that \(C_{d,n}\subseteq \mathcal{V}_{d,n}\subseteq \overline{C_{d,n}}\) and \(\mathcal{V}_{d,n}\) is a convex cone in the space of \(n\)-variate homogeneous polynomials of degree \(d\). In Theorem 1.4, they provide a complete characterization of binary forms whose sublevel sets have finite Lebesgue volume: for \(g\in \overline{C_{d,2}}\), \(g\in\mathcal{V}_{d,2}\) if and only if \(g\) has a zero of order at most \(\frac{d}{2}-1\). In particular, \(\mathcal{V}_{4,2} = \mathcal{C}_{4,2}\).
Moreover, the authors prove that sublevel sets have finite volume for general polynomials on the boundary of the cone of nonnegative forms (see Theorem 1.6). Finally, they solve the problem of comparing \(L^2\) and \(L^1\) norms of the measure naturally associated with a positive definite form (see Theorem 1.12).