{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:V6VDX5U25GHE3MYYQO47LJTZ2K","short_pith_number":"pith:V6VDX5U2","schema_version":"1.0","canonical_sha256":"afaa3bf69ae98e4db31883b9f5a679d2bc9eea69a8e5283f79de16fcbab88962","source":{"kind":"arxiv","id":"1605.02042","version":3},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.02042","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-06T19:12:04Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"fa4984a6b988b45abb1b48269d11a19279a377e16061e80196bc3f231c11ab53","abstract_canon_sha256":"9433e92b46a819b0a2b6d6fa3d7f5937dba636d2bfcde8a20f723edd9e54c686"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:34.471882Z","signature_b64":"sT+3fukPzsLZyOZvyD0YBxx1l9dzUf2pzfiWM+SoNLxZyxP0paVOYEkkK5XFR+CragJH89Iqdp6rZzC/ktxJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afaa3bf69ae98e4db31883b9f5a679d2bc9eea69a8e5283f79de16fcbab88962","last_reissued_at":"2026-05-18T00:59:34.471386Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:34.471386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.02042","created_at":"2026-05-18T00:59:34.471468+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.02042v3","created_at":"2026-05-18T00:59:34.471468+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02042","created_at":"2026-05-18T00:59:34.471468+00:00"},{"alias_kind":"pith_short_12","alias_value":"V6VDX5U25GHE","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"V6VDX5U25GHE3MYY","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"V6VDX5U2","created_at":"2026-05-18T12:30:48.956258+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K","json":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K.json","graph_json":"https://pith.science/api/pith-number/V6VDX5U25GHE3MYYQO47LJTZ2K/graph.json","events_json":"https://pith.science/api/pith-number/V6VDX5U25GHE3MYYQO47LJTZ2K/events.json","paper":"https://pith.science/paper/V6VDX5U2"},"agent_actions":{"view_html":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K","download_json":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K.json","view_paper":"https://pith.science/paper/V6VDX5U2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.02042&json=true","fetch_graph":"https://pith.science/api/pith-number/V6VDX5U25GHE3MYYQO47LJTZ2K/graph.json","fetch_events":"https://pith.science/api/pith-number/V6VDX5U25GHE3MYYQO47LJTZ2K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K/action/storage_attestation","attest_author":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K/action/author_attestation","sign_citation":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K/action/citation_signature","submit_replication":"https://pith.science/pith/V6VDX5U25GHE3MYYQO47LJTZ2K/action/replication_record"}},"created_at":"2026-05-18T00:59:34.471468+00:00","updated_at":"2026-05-18T00:59:34.471468+00:00"}