{"paper":{"title":"A Jordan-like decomposition theorem for valuations on star bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Ignacio Villanueva, Pedro Tradacete","submitted_at":"2016-05-06T19:12:04Z","abstract_excerpt":"We show that every radial continuous valuation $V:\\mathcal S_0^n\\rightarrow \\mathbb R$ defined on the $n$-dimensional star bodies $\\mathcal S_0^n$, and verifying $V(\\{0\\})=0$, can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations on $\\mathcal S_0^n$ with $V^+(\\{0\\})=V^-(\\{0\\})=0$.\n  As an application, we show that radial continuous rotationally invariant valuations $V$ on $\\mathcal S_0^n$ can be characterized as the applications on star bodies which can be written as $$V(K)=\\int_{S^{n-1}}\\theta(\\rho_K)dm,$$ where $\\theta:[0,\\infty)\\rightar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02042","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}