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Article Dans Une Revue SIAM Journal on Applied Algebra and Geometry Année : 2023

Volumes of sublevel sets of nonnegative forms and complete monotonicity

Résumé

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.
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Dates et versions

hal-03693810 , version 1 (13-06-2022)

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Khazhgali Kozhasov, Jean-Bernard Lasserre. Volumes of sublevel sets of nonnegative forms and complete monotonicity. SIAM Journal on Applied Algebra and Geometry, 2023, 7 (4), ⟨10.1137/22M1502458⟩. ⟨hal-03693810⟩
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