The Inverse Moment Problem for Convex Polytopes

We present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, assuming knowledge of finitely many of its integral moments. In particular, we show that the vertices of an N-vertex convex polytope in ℝd can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure of degree D), in d+1 distinct directions in general position. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii–Pukhlikov, and Barvinok that arise in the discrete geometry of polytopes, combined with what is variously known as Prony’s method, or the Vandermonde factorization of finite rank Hankel matrices.