Recovering an Homogeneous Polynomial from Moments of Its Level Set
Let be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial . Assume that the only knowledge about is the degree of as well as the moments of the Lebesgue measure on up to order . Then the vector of coefficients of is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order of the Lebesgue measure on encode all information on the homogeneous polynomial that defines (in fact, only moments of order and are needed).
homogeneous polynomials moments sublevel sets inverse problem from moments