{"paper":{"title":"Projections of convex polytopes to a line and higher univariate Prony systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The directional moment variety of convex polytope projections equals the Hankel determinantal variety of measures on polytopes.","cross_cats":["math.AG"],"primary_cat":"math.CA","authors_text":"Boris Shapiro","submitted_at":"2026-05-15T12:56:20Z","abstract_excerpt":"Motivated by the inverse moment problem for convex polytopes, we study the pushforward to a line of the Lebesgue measure restricted to a convex $d$-polytope. Such pushforwards are spline densities of degree $d-1$, and their moments lead naturally to a family of ``higher'' univariate Prony systems, with the classical Prony system recovered when $d=0$. We describe the corresponding fixed-knot spline cone, give an explicit amplitude recovery criterion, record the rational generating function and recurrence satisfied by the normalized moments, and identify the directional moment variety with the H"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We identify the directional moment variety with the Hankel determinantal variety appearing in the theory of moment varieties of measures on polytopes.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The pushforward to a line of the Lebesgue measure restricted to a convex d-polytope is a spline density of degree d-1 whose moments lead naturally to a family of higher univariate Prony systems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Pushforwards of Lebesgue measure restricted to convex d-polytopes onto a line produce spline densities whose moments obey higher Prony systems, with the directional moment variety identified as the Hankel determinantal variety from moment varieties of measures on polytopes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The directional moment variety of convex polytope projections equals the Hankel determinantal variety of measures on polytopes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"14c97d9feb636f1adebd73a46f1366a5d6ac4aa5aa48f0a5bf000fa564d736c6"},"source":{"id":"2605.15917","kind":"arxiv","version":1},"verdict":{"id":"177b30a5-0b0b-4742-8ab8-89a0a92fd701","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:42:41.033368Z","strongest_claim":"We identify the directional moment variety with the Hankel determinantal variety appearing in the theory of moment varieties of measures on polytopes.","one_line_summary":"Pushforwards of Lebesgue measure restricted to convex d-polytopes onto a line produce spline densities whose moments obey higher Prony systems, with the directional moment variety identified as the Hankel determinantal variety from moment varieties of measures on polytopes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The pushforward to a line of the Lebesgue measure restricted to a convex d-polytope is a spline density of degree d-1 whose moments lead naturally to a family of higher univariate Prony systems.","pith_extraction_headline":"The directional moment variety of convex polytope projections equals the Hankel determinantal variety of measures on polytopes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15917/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T18:01:18.552948Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T17:50:27.509768Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:46.554474Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.755658Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"98f40f61aff433b1110564c544f3a3fd54434f74793e3281492acbefbd7472fd"},"references":{"count":11,"sample":[{"doi":"","year":2018,"title":"C. Am´ endola, K. Ranestad, and B. Sturmfels,Algebraic identifiability of Gaussian mixtures, Int. Math. Res. Not. IMRN 2018, no. 21, 6551–6566","work_id":"30f78a35-e1c4-4b0b-aa20-db945f35dad1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1966,"title":"H. B. Curry and I. J. Schoenberg,On P´ olya frequency functions. IV. The fundamental spline functions and their limits, J. Analyse Math.17(1966), 71–107","work_id":"bcd52d69-b110-4b67-b902-f724bd8ba557","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"De Concini and C","work_id":"d3ed371c-8556-46d9-a53a-43b0dd9ef680","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"R. J. Gardner,Geometric Tomography, second edition, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 2006","work_id":"3ea80c60-67b0-4ce2-80c5-69af015b7356","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"N. Gravin, J. B. Lasserre, D. V. Pasechnik, and S. Robins,The inverse moment problem for convex polytopes, Discrete Comput. Geom.48(2012), 596–621","work_id":"6c3e87cf-8242-4eef-9967-fae149153e4f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"8aa4629426254166c4f55fdb858d5cd7df07f86f105e88cdad007610f1e02f48","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}