Level sets and non Gaussian integrals of positively homogeneous functions

Details Level sets and non Gaussian integrals of positively homogeneous functions Level sets and non Gaussian integrals of positively homogeneous functions

2011-10-30

arxiv.org/abs/1110.6632v3

Mathematics

Optimization and Control

Scientific paper

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int \phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $\phi:R_+\to R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

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