Volume of sub-level sets of polynomials

Consider the sub-level set $\mathbf{K}:=\{\mathrm{x}:g(\mathrm{x})\leq 1\}$ of a nonnegative homogeneous polynomial $g$. We show that its Lebesgue volume vol(K) can be approximated as closely as desired by solving a sequence of generalized eigenvalue problems with respect to a pair of Hankel matrices of increasing size, whose entries are obtained in closed form. An extension to the non-homogeneous case is also briefly described.