{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:V6VDX5U25GHE3MYYQO47LJTZ2K","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9433e92b46a819b0a2b6d6fa3d7f5937dba636d2bfcde8a20f723edd9e54c686","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-06T19:12:04Z","title_canon_sha256":"fa4984a6b988b45abb1b48269d11a19279a377e16061e80196bc3f231c11ab53"},"schema_version":"1.0","source":{"id":"1605.02042","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02042","created_at":"2026-05-18T00:59:34Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02042v3","created_at":"2026-05-18T00:59:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02042","created_at":"2026-05-18T00:59:34Z"},{"alias_kind":"pith_short_12","alias_value":"V6VDX5U25GHE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"V6VDX5U25GHE3MYY","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"V6VDX5U2","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:ddbbf6b09f172ed00d2c2ae8b2e5b3c37fd18d4e21f5bc3101468cb62de2c980","target":"graph","created_at":"2026-05-18T00:59:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Ignacio Villanueva, Pedro Tradacete","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-06T19:12:04Z","title":"A Jordan-like decomposition theorem for valuations on star bodies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02042","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f630533d39ae34d43ec36bf99d4a1e0589f7259c392a7a0aacdcd721e25dad37","target":"record","created_at":"2026-05-18T00:59:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9433e92b46a819b0a2b6d6fa3d7f5937dba636d2bfcde8a20f723edd9e54c686","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-06T19:12:04Z","title_canon_sha256":"fa4984a6b988b45abb1b48269d11a19279a377e16061e80196bc3f231c11ab53"},"schema_version":"1.0","source":{"id":"1605.02042","kind":"arxiv","version":3}},"canonical_sha256":"afaa3bf69ae98e4db31883b9f5a679d2bc9eea69a8e5283f79de16fcbab88962","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"afaa3bf69ae98e4db31883b9f5a679d2bc9eea69a8e5283f79de16fcbab88962","first_computed_at":"2026-05-18T00:59:34.471386Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:34.471386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sT+3fukPzsLZyOZvyD0YBxx1l9dzUf2pzfiWM+SoNLxZyxP0paVOYEkkK5XFR+CragJH89Iqdp6rZzC/ktxJBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:34.471882Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.02042","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f630533d39ae34d43ec36bf99d4a1e0589f7259c392a7a0aacdcd721e25dad37","sha256:ddbbf6b09f172ed00d2c2ae8b2e5b3c37fd18d4e21f5bc3101468cb62de2c980"],"state_sha256":"ef368d57e602a638c680c12e8b7436eeb9d12b32aeca4b37dba60dc2d9969bc4"}