Volume of Sublevel Sets of Homogeneous Polynomials

Consider the sublevel set $K:=\{x: g(x)\leq1\}$ 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. The methodology also extends to compact sets of the form $\{x: a\leq g(x)\leq b\}$ for nonhomogeneous polynomials with degree $d\ll n$. It reduces the volume computation in $\mathbb {R}^n$ to a “volume" computation in $\mathbb {R}^d$ (where $d={deg}(g)$) for a certain pushforward measure. Another extension to computing volumes of finite intersections of such sublevel sets is also briefly described.