{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:5BTKRSB27TF24UBPRSEMQTS44E","short_pith_number":"pith:5BTKRSB2","canonical_record":{"source":{"id":"1207.7256","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-31T14:23:07Z","cross_cats_sorted":["math.DG","math.FA"],"title_canon_sha256":"a33dda8c44d40127fd15a15e7e77a2b808503a946b70a08388623281357a4c82","abstract_canon_sha256":"1890438a1510938919ffa00273cc2ff0e5a997e2830b6b97ffe5445f0ef66574"},"schema_version":"1.0"},"canonical_sha256":"e866a8c83afccbae502f8c88c84e5ce11075cc7ed308427c0a85ee5848255944","source":{"kind":"arxiv","id":"1207.7256","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7256","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7256v1","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7256","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"pith_short_12","alias_value":"5BTKRSB27TF2","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"5BTKRSB27TF24UBP","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"5BTKRSB2","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:5BTKRSB27TF24UBPRSEMQTS44E","target":"record","payload":{"canonical_record":{"source":{"id":"1207.7256","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-31T14:23:07Z","cross_cats_sorted":["math.DG","math.FA"],"title_canon_sha256":"a33dda8c44d40127fd15a15e7e77a2b808503a946b70a08388623281357a4c82","abstract_canon_sha256":"1890438a1510938919ffa00273cc2ff0e5a997e2830b6b97ffe5445f0ef66574"},"schema_version":"1.0"},"canonical_sha256":"e866a8c83afccbae502f8c88c84e5ce11075cc7ed308427c0a85ee5848255944","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:46.386657Z","signature_b64":"ngBMgVsrt4oGN/VQ0wbgjBgpTfOq+KXvo/0ZUQ70TAjNiGNrZvCrZCueYijA6EtqcRBgcVXhTXSVwggXfIynCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e866a8c83afccbae502f8c88c84e5ce11075cc7ed308427c0a85ee5848255944","last_reissued_at":"2026-05-18T03:49:46.385719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:46.385719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.7256","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8HjcgxGJBKhzzzmXCJwwAovdbitXJqVLof2DCF/5xotsDIdZnpMZ4p2r6j4b8H8by/+Ohcpj7N9KGAcE6G7TAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T11:30:37.453121Z"},"content_sha256":"dc08d6904439fcf528202f6dc5d018ecaa10a4db60866b58f62560e6e024bffd","schema_version":"1.0","event_id":"sha256:dc08d6904439fcf528202f6dc5d018ecaa10a4db60866b58f62560e6e024bffd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:5BTKRSB27TF24UBPRSEMQTS44E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"GL(n) contravariant Minkowski valuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA"],"primary_cat":"math.MG","authors_text":"Franz E. Schuster, Thomas Wannerer","submitted_at":"2012-07-31T14:23:07Z","abstract_excerpt":"A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7256","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C7YKYMyiPxVxRPTeLQRZYLT4ac8eD4I6gRfSbJTaag47raUptcIMq+396AzN11OjGUgYf4JBjliQTaaaySblAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T11:30:37.453926Z"},"content_sha256":"de57c1f6c0241c5db86fda12bb4666a3b9aead1e056f2d289c51aca16e4c02aa","schema_version":"1.0","event_id":"sha256:de57c1f6c0241c5db86fda12bb4666a3b9aead1e056f2d289c51aca16e4c02aa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5BTKRSB27TF24UBPRSEMQTS44E/bundle.json","state_url":"https://pith.science/pith/5BTKRSB27TF24UBPRSEMQTS44E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5BTKRSB27TF24UBPRSEMQTS44E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-19T11:30:37Z","links":{"resolver":"https://pith.science/pith/5BTKRSB27TF24UBPRSEMQTS44E","bundle":"https://pith.science/pith/5BTKRSB27TF24UBPRSEMQTS44E/bundle.json","state":"https://pith.science/pith/5BTKRSB27TF24UBPRSEMQTS44E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5BTKRSB27TF24UBPRSEMQTS44E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:5BTKRSB27TF24UBPRSEMQTS44E","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":"1890438a1510938919ffa00273cc2ff0e5a997e2830b6b97ffe5445f0ef66574","cross_cats_sorted":["math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-31T14:23:07Z","title_canon_sha256":"a33dda8c44d40127fd15a15e7e77a2b808503a946b70a08388623281357a4c82"},"schema_version":"1.0","source":{"id":"1207.7256","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7256","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7256v1","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7256","created_at":"2026-05-18T03:49:46Z"},{"alias_kind":"pith_short_12","alias_value":"5BTKRSB27TF2","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"5BTKRSB27TF24UBP","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"5BTKRSB2","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:de57c1f6c0241c5db86fda12bb4666a3b9aead1e056f2d289c51aca16e4c02aa","target":"graph","created_at":"2026-05-18T03:49:46Z","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":"A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.","authors_text":"Franz E. Schuster, Thomas Wannerer","cross_cats":["math.DG","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-31T14:23:07Z","title":"GL(n) contravariant Minkowski valuations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7256","kind":"arxiv","version":1},"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:dc08d6904439fcf528202f6dc5d018ecaa10a4db60866b58f62560e6e024bffd","target":"record","created_at":"2026-05-18T03:49:46Z","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":"1890438a1510938919ffa00273cc2ff0e5a997e2830b6b97ffe5445f0ef66574","cross_cats_sorted":["math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-31T14:23:07Z","title_canon_sha256":"a33dda8c44d40127fd15a15e7e77a2b808503a946b70a08388623281357a4c82"},"schema_version":"1.0","source":{"id":"1207.7256","kind":"arxiv","version":1}},"canonical_sha256":"e866a8c83afccbae502f8c88c84e5ce11075cc7ed308427c0a85ee5848255944","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e866a8c83afccbae502f8c88c84e5ce11075cc7ed308427c0a85ee5848255944","first_computed_at":"2026-05-18T03:49:46.385719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:46.385719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ngBMgVsrt4oGN/VQ0wbgjBgpTfOq+KXvo/0ZUQ70TAjNiGNrZvCrZCueYijA6EtqcRBgcVXhTXSVwggXfIynCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:46.386657Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.7256","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc08d6904439fcf528202f6dc5d018ecaa10a4db60866b58f62560e6e024bffd","sha256:de57c1f6c0241c5db86fda12bb4666a3b9aead1e056f2d289c51aca16e4c02aa"],"state_sha256":"02bba809dcd6e107be3676e8798cac48f56f0c2df468acdbc9cea3d440afec86"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xg60PRsyaWCZJo3qXIRTeA/XUHMsIHvSmZpjbP4gi36ShNn8VlQfcG4zxAICEd0IWm5WKA2B8KHnOALYg73FCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T11:30:37.457239Z","bundle_sha256":"7b8efc032d653072c64935758dbea1c6f357496530cfc4b555f65f0d6bbf6991"}}