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

2022/06/10 by Khazhgali Kozhasov, Kozhasov, Khazhgali, Jean B. Lasserre +1
Mathematics · #14P10 #26B15 #26C05 #26C10 #44A10 #51M25 #90C23 #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Morphological variations and asymmetry #Optimization and Control (math.OC) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2206.05170

openalex publication_date 2022/06/10 · openalex created_date 2022/06/14 · openalex updated_date 2026/07/28

Abstract

Let C_d,n be the convex cone consisting of real n-variate degree d forms that are strictly positive on ℝⁿ∖ \0\. We prove that the Lebesgue volume of the sublevel set \g≤ 1\ of g∈ C_d,n is a completely monotone function on 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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