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 d can be reconstructed from the knowledge of O(DN) axial m...

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Bibliographic Details
Main Authors: Gravin, Nick., Lasserre, Jean., Pasechnik, Dmitrii V., Robins, Sinai.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2013
Online Access:https://hdl.handle.net/10356/95885
http://hdl.handle.net/10220/10865
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Institution: Nanyang Technological University
Language: English
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Summary: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 d 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.