Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

Jean B. Lasserre ()
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Jean B. Lasserre: LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS 7 Avenue du Colonel Roche, BP 54200, 31031 Toulouse Cedex 4-France, France

International Game Theory Review (IGTR), 2015, vol. 17, issue 01, 1-28

Abstract: We investigate various properties of the sublevel setG= {x: g(x) ≤ 1}and the integration ofhon this sublevel set whengandhare positively homogeneous functions (and in particular homogeneous polynomials). For instance, the latter integral reduces to integratinghexp(-g)on the whole spaceℝn(a nonGaussian integral) and whengis a polynomial, then the volume ofGis a convex function of the coefficients ofg. We also provide a numerical approximation scheme to compute the volume ofGor integratehonG(or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set{x: g(x) ≤ 1}of minimum volume that contains some given subsetKis a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

Keywords: Positively homogeneous functions; nonGaussian integrals; volume; convex optimization; 26B15; 65K10; 90C22; 90C25 (search for similar items in EconPapers)
JEL-codes: B4 C0 C6 C7 D5 D7 M2 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1142/S0219198915400010

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