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

Khazhgali Kozhasov; Jean B. Lasserre

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arXiv · 2206.05170

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

Khazhgali Kozhasov·Jean B. Lasserre

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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.

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BibTeX RIS

Khazhgali Kozhasov, Jean B. Lasserre. 2022-06-10. Non-negative forms, volumes of sublevel sets, complete monotonicity and moment matrices. https://arxiv.org/abs/2206.05170

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