We investigate various properties of the sublevel set {x:g(x)β€1}
and the integration of h on this sublevel set when g and hare positively
homogeneous functions. For instance, the latter integral reduces to integrating
hexp(βg) on the whole space Rn (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 β«Ο(g(x))h(x)dx is a convex function of g for a large class of functions
Ο:R+ββR. 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)β€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