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 R 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-Pukhikov, 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.