Recovering an Homogeneous Polynomial from Moments of Its Level Set

Abstract

Let K := {x: g(x)≤ 1} be the compact (and not necessarily convex) 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 the 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). © 2013 Springer Science+Business Media New York.