  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.
