The inverse moment problem for convex polytopes

Abstract

The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. 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, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.

LaTeX2e, 24 pages including 1 appendix

References in corpus (2)

• How to Integrate a Polynomial over a Simplex
• Polynomial-time computing over quadratic maps I: sampling in real algebraic sets

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