On moments of a polytope
We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P ⊂ Rd is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to...
| Main Authors: | , , , |
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| Format: | Journal article |
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Springer International Publishing
2018
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| _version_ | 1797061150712528896 |
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| author | Gravin, N Pasechnik, D Shapiro, B Shapiro, M |
| author_facet | Gravin, N Pasechnik, D Shapiro, B Shapiro, M |
| author_sort | Gravin, N |
| collection | OXFORD |
| description | We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P ⊂ Rd is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S. |
| first_indexed | 2024-03-06T20:26:57Z |
| format | Journal article |
| id | oxford-uuid:2fb84b81-caa1-4a12-891a-e4614200690c |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:26:57Z |
| publishDate | 2018 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | oxford-uuid:2fb84b81-caa1-4a12-891a-e4614200690c2022-03-26T12:57:01ZOn moments of a polytopeJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2fb84b81-caa1-4a12-891a-e4614200690cSymplectic Elements at OxfordSpringer International Publishing2018Gravin, NPasechnik, DShapiro, BShapiro, MWe show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P ⊂ Rd is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S. |
| spellingShingle | Gravin, N Pasechnik, D Shapiro, B Shapiro, M On moments of a polytope |
| title | On moments of a polytope |
| title_full | On moments of a polytope |
| title_fullStr | On moments of a polytope |
| title_full_unstemmed | On moments of a polytope |
| title_short | On moments of a polytope |
| title_sort | on moments of a polytope |
| work_keys_str_mv | AT gravinn onmomentsofapolytope AT pasechnikd onmomentsofapolytope AT shapirob onmomentsofapolytope AT shapirom onmomentsofapolytope |