Level Sets and NonGaussian Integrals of Positively Homogeneous Functions

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.