The inverse moment problem for convex polytopes
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....
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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2011
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| _version_ | 1797071178607624192 |
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| author | Gravin, N Lasserre, J Pasechnik, D Robins, S |
| author_facet | Gravin, N Lasserre, J Pasechnik, D Robins, S |
| author_sort | Gravin, N |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T22:49:32Z |
| format | Journal article |
| id | oxford-uuid:5e549279-492a-450f-9970-4df5b31d4f89 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:49:32Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:5e549279-492a-450f-9970-4df5b31d4f892022-03-26T17:39:59ZThe inverse moment problem for convex polytopesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5e549279-492a-450f-9970-4df5b31d4f89EnglishSymplectic Elements at Oxford2011Gravin, NLasserre, JPasechnik, DRobins, SThe 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. |
| spellingShingle | Gravin, N Lasserre, J Pasechnik, D Robins, S The inverse moment problem for convex polytopes |
| title | 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_short | The inverse moment problem for convex polytopes |
| title_sort | inverse moment problem for convex polytopes |
| work_keys_str_mv | AT gravinn theinversemomentproblemforconvexpolytopes AT lasserrej theinversemomentproblemforconvexpolytopes AT pasechnikd theinversemomentproblemforconvexpolytopes AT robinss theinversemomentproblemforconvexpolytopes AT gravinn inversemomentproblemforconvexpolytopes AT lasserrej inversemomentproblemforconvexpolytopes AT pasechnikd inversemomentproblemforconvexpolytopes AT robinss inversemomentproblemforconvexpolytopes |