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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sg-ntu-dr.10356-958852020-03-07T12:37:22Z The inverse moment problem for convex polytopes Gravin, Nick. Lasserre, Jean. Pasechnik, Dmitrii V. Robins, Sinai. School of Physical and Mathematical Sciences 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. 2013-07-01T07:00:28Z 2019-12-06T19:22:51Z 2013-07-01T07:00:28Z 2019-12-06T19:22:51Z 2012 2012 Journal Article Gravin, N., Lasserre, J., Pasechnik, D. V., & Robins, S. (2012). The Inverse Moment Problem for Convex Polytopes. Discrete & Computational Geometry, 48(3), 596-621. 0179-5376 https://hdl.handle.net/10356/95885 http://hdl.handle.net/10220/10865 10.1007/s00454-012-9426-4 en Discrete & computational geometry © 2012 Springer Science+Business Media, LLC. |
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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. |
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School of Physical and Mathematical Sciences |
| author_facet |
School of Physical and Mathematical Sciences Gravin, Nick. Lasserre, Jean. Pasechnik, Dmitrii V. Robins, Sinai. |
| format |
Article |
| author |
Gravin, Nick. Lasserre, Jean. Pasechnik, Dmitrii V. Robins, Sinai. |
| spellingShingle |
Gravin, Nick. Lasserre, Jean. Pasechnik, Dmitrii V. Robins, Sinai. The inverse moment problem for convex polytopes |
| author_sort |
Gravin, Nick. |
| title |
The inverse moment problem for convex polytopes |
| title_short |
The inverse moment problem for convex polytopes |
| title_full |
The inverse moment problem for convex polytopes |
| title_fullStr |
The inverse moment problem for convex polytopes |
| title_full_unstemmed |
The inverse moment problem for convex polytopes |
| title_sort |
inverse moment problem for convex polytopes |
| publishDate |
2013 |
| url |
https://hdl.handle.net/10356/95885 http://hdl.handle.net/10220/10865 |
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1681045126826164224 |