{"paper":{"title":"The Dual Minkowski Problem under Group Actions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces.","cross_cats":[],"primary_cat":"math.MG","authors_text":"Junjie Shan","submitted_at":"2026-05-15T12:21:59Z","abstract_excerpt":"In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies, recovering the origin-symmetric setting when $G=\\{\\pm I\\}$. The necessary and sufficient conditions concern the concentration of the measure on $G$-invariant subspaces, both in the range $0<q<n$ and at the critical endpoint $q=n$, where the problem becomes the logarithmic Minkowski problem."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For 0<q≤n, we give a complete existence characterization in the framework of G-invariant convex bodies, recovering the origin-symmetric setting when G={±I}. The necessary and sufficient conditions concern the concentration of the measure on G-invariant subspaces, both in the range 0<q<n and at the critical endpoint q=n.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The given measure must obey specific concentration restrictions on the G-invariant subspaces; if this concentration condition fails, no G-invariant solution exists, as this forms the necessary and sufficient criterion stated for both the subcritical and critical cases.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper establishes necessary and sufficient conditions for the existence of G-invariant convex bodies solving the dual Minkowski problem, with the conditions depending on measure concentration on G-invariant subspaces, including the logarithmic case at q = n.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"658b026a43e6b32055c09d8a64ab818c4c0b1ebf082d49ef0abfcf9271b2e8c8"},"source":{"id":"2605.15891","kind":"arxiv","version":1},"verdict":{"id":"8c50d871-cf3f-4ade-89f4-294fe3b2b9d1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:24:34.779056Z","strongest_claim":"For 0<q≤n, we give a complete existence characterization in the framework of G-invariant convex bodies, recovering the origin-symmetric setting when G={±I}. The necessary and sufficient conditions concern the concentration of the measure on G-invariant subspaces, both in the range 0<q<n and at the critical endpoint q=n.","one_line_summary":"The paper establishes necessary and sufficient conditions for the existence of G-invariant convex bodies solving the dual Minkowski problem, with the conditions depending on measure concentration on G-invariant subspaces, including the logarithmic case at q = n.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The given measure must obey specific concentration restrictions on the G-invariant subspaces; if this concentration condition fails, no G-invariant solution exists, as this forms the necessary and sufficient criterion stated for both the subcritical and critical cases.","pith_extraction_headline":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15891/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T17:36:25.372690Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:47.518818Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:18.441954Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.783361Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"2ba631560c0b873825f261966b51f4fa73522d596f7156ab9a9a2db1c747e1d7"},"references":{"count":46,"sample":[{"doi":"","year":2013,"title":"B¨ or¨ oczky, E","work_id":"12909b98-f0eb-42c0-b397-fba5844c8784","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"B¨ or¨ oczky, F","work_id":"6408a013-364f-4306-8d95-6e6ee6a291cf","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"B¨ or¨ oczky, P","work_id":"61cddad1-986d-4241-ab97-70f939d19466","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"B¨ or¨ oczky, M","work_id":"e0ef2f07-2eb8-4c2b-8916-00aa6b7ba221","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"B¨ or¨ oczky, M","work_id":"f1577371-a099-4392-ba54-781cc8f93d8b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":46,"snapshot_sha256":"fa80c2cfd3704ade4efc7ebfde9ebbd2a4964827764691bb5a6f7d7371231c6d","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"3363e61c38af867d481e7ab638f9a92b12e58d8093f3625d48a4dd10c9b5e069"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}