{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:OEJMPNEFUT4HOL4S7X5LHIW3WB","short_pith_number":"pith:OEJMPNEF","canonical_record":{"source":{"id":"2210.02359","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-10-05T16:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"3a0526a1fee5045906f143d56157287c82456ecedab17dddcae5ee97628448df","abstract_canon_sha256":"65e478fabb6a7ae01d3cb8132d66656e41cf0472d66491461b536aff8ea03c22"},"schema_version":"1.0"},"canonical_sha256":"7112c7b485a4f8772f92fdfab3a2dbb04c677ff6fbecc041108c15f938f60134","source":{"kind":"arxiv","id":"2210.02359","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.02359","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"arxiv_version","alias_value":"2210.02359v2","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.02359","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_12","alias_value":"OEJMPNEFUT4H","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_16","alias_value":"OEJMPNEFUT4HOL4S","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_8","alias_value":"OEJMPNEF","created_at":"2026-07-05T06:50:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:OEJMPNEFUT4HOL4S7X5LHIW3WB","target":"record","payload":{"canonical_record":{"source":{"id":"2210.02359","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-10-05T16:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"3a0526a1fee5045906f143d56157287c82456ecedab17dddcae5ee97628448df","abstract_canon_sha256":"65e478fabb6a7ae01d3cb8132d66656e41cf0472d66491461b536aff8ea03c22"},"schema_version":"1.0"},"canonical_sha256":"7112c7b485a4f8772f92fdfab3a2dbb04c677ff6fbecc041108c15f938f60134","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:50:27.549203Z","signature_b64":"NYNywlszCesCZZ2oM0Fo2i+GTWLzKYGxOdndFzxOe2lTw0xTviNDIgnb+6AMm2X074nQz+00eU/zRjFAon2jAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7112c7b485a4f8772f92fdfab3a2dbb04c677ff6fbecc041108c15f938f60134","last_reissued_at":"2026-07-05T06:50:27.548739Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:50:27.548739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2210.02359","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:50:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tb2BG73VslgfNE+9C2YZ7GacGU0TW9deedd9MMu4cgs2rHej/zDsrkrGWRjuyUckiZrjUB2W/xqi0ZOMrWATAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T17:07:10.189440Z"},"content_sha256":"fb862c5cd291cdc6cd2934acd4ab451a97f5f641d05fbf217ff7beb92bcfa7cd","schema_version":"1.0","event_id":"sha256:fb862c5cd291cdc6cd2934acd4ab451a97f5f641d05fbf217ff7beb92bcfa7cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:OEJMPNEFUT4HOL4S7X5LHIW3WB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dual curvature measures for log-concave functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Dongmeng Xi, Jiaqian Liu, Yiming Zhao, Yong Huang","submitted_at":"2022-10-05T16:03:28Z","abstract_excerpt":"We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are shown. The associated Minkowski problem for these dual curvature measures is considered and sufficient conditions in the symmetric setting are demonstrated."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.02359","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.02359/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:50:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U6N/9jr6FLeK1lY8QUE8jXIO1zQ5nYO09wf1h0cwmshOijs2ZPjgZeVI/oRK9/+XX5/oUK+ApTOkVHJXjDT8CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T17:07:10.189822Z"},"content_sha256":"2ffce689e36e346aaad39ed5bbbda2ed0f5b0e5be4bb6509bd803fed730b4fae","schema_version":"1.0","event_id":"sha256:2ffce689e36e346aaad39ed5bbbda2ed0f5b0e5be4bb6509bd803fed730b4fae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/bundle.json","state_url":"https://pith.science/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/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":"2026-08-13T17:07:10Z","links":{"resolver":"https://pith.science/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB","bundle":"https://pith.science/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/bundle.json","state":"https://pith.science/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OEJMPNEFUT4HOL4S7X5LHIW3WB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:OEJMPNEFUT4HOL4S7X5LHIW3WB","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":"65e478fabb6a7ae01d3cb8132d66656e41cf0472d66491461b536aff8ea03c22","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-10-05T16:03:28Z","title_canon_sha256":"3a0526a1fee5045906f143d56157287c82456ecedab17dddcae5ee97628448df"},"schema_version":"1.0","source":{"id":"2210.02359","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.02359","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"arxiv_version","alias_value":"2210.02359v2","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.02359","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_12","alias_value":"OEJMPNEFUT4H","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_16","alias_value":"OEJMPNEFUT4HOL4S","created_at":"2026-07-05T06:50:27Z"},{"alias_kind":"pith_short_8","alias_value":"OEJMPNEF","created_at":"2026-07-05T06:50:27Z"}],"graph_snapshots":[{"event_id":"sha256:2ffce689e36e346aaad39ed5bbbda2ed0f5b0e5be4bb6509bd803fed730b4fae","target":"graph","created_at":"2026-07-05T06:50:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2210.02359/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are shown. The associated Minkowski problem for these dual curvature measures is considered and sufficient conditions in the symmetric setting are demonstrated.","authors_text":"Dongmeng Xi, Jiaqian Liu, Yiming Zhao, Yong Huang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-10-05T16:03:28Z","title":"Dual curvature measures for log-concave functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.02359","kind":"arxiv","version":2},"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:fb862c5cd291cdc6cd2934acd4ab451a97f5f641d05fbf217ff7beb92bcfa7cd","target":"record","created_at":"2026-07-05T06:50:27Z","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":"65e478fabb6a7ae01d3cb8132d66656e41cf0472d66491461b536aff8ea03c22","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-10-05T16:03:28Z","title_canon_sha256":"3a0526a1fee5045906f143d56157287c82456ecedab17dddcae5ee97628448df"},"schema_version":"1.0","source":{"id":"2210.02359","kind":"arxiv","version":2}},"canonical_sha256":"7112c7b485a4f8772f92fdfab3a2dbb04c677ff6fbecc041108c15f938f60134","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7112c7b485a4f8772f92fdfab3a2dbb04c677ff6fbecc041108c15f938f60134","first_computed_at":"2026-07-05T06:50:27.548739Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:50:27.548739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NYNywlszCesCZZ2oM0Fo2i+GTWLzKYGxOdndFzxOe2lTw0xTviNDIgnb+6AMm2X074nQz+00eU/zRjFAon2jAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:50:27.549203Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.02359","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb862c5cd291cdc6cd2934acd4ab451a97f5f641d05fbf217ff7beb92bcfa7cd","sha256:2ffce689e36e346aaad39ed5bbbda2ed0f5b0e5be4bb6509bd803fed730b4fae"],"state_sha256":"8741d4446d0bd73f167f2439a0175ee7b71c7d5ca8ebd3b5677b9a9567376231"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IrIB6pvHztcpAxJsK5ygBTU8EeoOp12FMxsD0XV46vtzB1YnYs86CCGVrT8i46wBiEQ8CZkGO90s5iRf1YCWBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T17:07:10.193363Z","bundle_sha256":"bbd73314094f4fbb2449a2316e49e7fcd15216fa90194a03898259802b928a6f"}}