 SIGMAOPT SEMINAR
 Speaker: Dr Jean Lasserre, LAASCNRS, Université de Toulouse
 Title: Sublevel sets of positively homogeneous functions and nonGaussian integrals
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 3:00 pm, Wed, 11^{th} Apr 2012
 (Rescheduled from 10th April)
 Abstract:
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 $\mathbb{R}^n$ (a nonGaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of its coefficients.
In fact, whenever $h$ is nonnegative, the functional $\int \phi(g)h dx$ is a convex function of $g$ for a large class of functions $\phi:\mathbb{R}_{+} \to \mathbb{R}$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently, to approximate the associated nonGaussian 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 Gaussianlike property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.
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