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arXiv · 1106.5723

The inverse moment problem for convex polytopes

Nick Gravin·Jean Lasserre·Dmitrii Pasechnik·Sinai Robins
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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.

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Nick Gravin, Jean Lasserre, Dmitrii Pasechnik, Sinai Robins. 2012-04-22. The inverse moment problem for convex polytopes. https://doi.org/10.1007/s00454-012-9426-4

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