Recovering an homogeneous polynomial from moments of its level set

Overview & Synopsis

Let $K:=\{x: g(x)\leq 1\}$ be the compact sub-level set of some homogeneous polynomial $g$. Assume that the only knowledge about $K$ is the degree of $g$ as well as the moments of the Lebesgue measure on $K$ up to order $2d$. Then the vector of coefficients of $g$ is solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $2d$ of the Lebesgue measure on $K$ encode all information on the homogeneous polynomial $g$ that defines $K$ (in fact, only moments of order $d$ and $2d$ are needed).