{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7Y3MK2VFON7XAPR6OKXI6ZR6EB","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":"d2c72c388be0264d5ea5a5b7f859f66545d88f12bb2f98d0e997bac2e4e6a03a","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-03-18T06:30:04Z","title_canon_sha256":"6d736d8d18bd4194220f48421388096d980dee0759258e3575df68748971377a"},"schema_version":"1.0","source":{"id":"1703.06259","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.06259","created_at":"2026-05-18T00:48:23Z"},{"alias_kind":"arxiv_version","alias_value":"1703.06259v1","created_at":"2026-05-18T00:48:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.06259","created_at":"2026-05-18T00:48:23Z"},{"alias_kind":"pith_short_12","alias_value":"7Y3MK2VFON7X","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Y3MK2VFON7XAPR6","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Y3MK2VF","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:990d75e84f4c8f4db1cc967684f779752567970db222ed8a354ab501cdd718fb","target":"graph","created_at":"2026-05-18T00:48:23Z","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":"The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\\mathbb{R}^n$. A full solution to this is given when $1 < q < n$.\n  The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved u","authors_text":"Deane Yang, Erwin Lutwak, Gaoyong Zhang, K\\'aroly B\\\"or\\\"oczky, Yiming Zhao","cross_cats":["math.AP","math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-03-18T06:30:04Z","title":"The dual Minkowski problem for symmetric convex bodies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06259","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":"sha256:43e45e6dd20f8994dcab0dab4048c619e9ff7698747037fa00677495eef2891e","target":"record","created_at":"2026-05-18T00:48:23Z","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":"d2c72c388be0264d5ea5a5b7f859f66545d88f12bb2f98d0e997bac2e4e6a03a","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-03-18T06:30:04Z","title_canon_sha256":"6d736d8d18bd4194220f48421388096d980dee0759258e3575df68748971377a"},"schema_version":"1.0","source":{"id":"1703.06259","kind":"arxiv","version":1}},"canonical_sha256":"fe36c56aa5737f703e3e72ae8f663e205da5292af4baaccb34f4c0930deff6b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe36c56aa5737f703e3e72ae8f663e205da5292af4baaccb34f4c0930deff6b2","first_computed_at":"2026-05-18T00:48:23.454774Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:23.454774Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PYLmrDiLJOuLPHb6j/XQp87/ssfA+CRk5WP4SimGrtlOFietk9gvVf5T5TDy6Bdu6YFUgUJW7QHTERyYDfSeCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:23.455356Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.06259","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43e45e6dd20f8994dcab0dab4048c619e9ff7698747037fa00677495eef2891e","sha256:990d75e84f4c8f4db1cc967684f779752567970db222ed8a354ab501cdd718fb"],"state_sha256":"be31082d0ecdfdbe8402d11419665e1e49a1089d6d8e20ce9fb71bcc05e736c6"}