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- Title
The Inverse Moment Problem for Convex Polytopes.
- Authors
Gravin, Nick; Lasserre, Jean; Pasechnik, Dmitrii; Robins, Sinai
- Abstract
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-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.
- Subjects
CONVEX polytopes; INVERSE problems; TOMOGRAPHY; MATHEMATICAL models of signal processing; X-ray microscopy; DISCRETE geometry; MATHEMATICAL models
- Publication
Discrete & Computational Geometry, 2012, Vol 48, Issue 3, p596
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-012-9426-4