{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:DNKA7EKNOI6ADAIHZ67QQ2RSUM","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":"bd2ccaa1abfea10ffe0f29fdb8ab1be55739b256bca8680f063e01e2ec4c913a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-02-18T00:21:09Z","title_canon_sha256":"6f7c1c50a59c2e07262b4ced1171e937e5065220775afead6ff91df9a971364e"},"schema_version":"1.0","source":{"id":"1202.4043","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4043","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4043v4","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4043","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"DNKA7EKNOI6A","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DNKA7EKNOI6ADAIH","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DNKA7EKN","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:f2b7afa75bb33d816529698fb0881019c0ec02cead94385ffea4dde52d6d188d","target":"graph","created_at":"2026-05-18T03:41:01Z","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 closed convex cone K is called nice, if the set K^* + F^\\perp is closed for all F faces of K, where K^* is the dual cone of K, and F^\\perp is the orthogonal complement of the linear span of F. The niceness property is important for two reasons: it plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer.\n  We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; in reverse, facial exposedn","authors_text":"Gabor Pataki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-02-18T00:21:09Z","title":"On the connection of facially exposed and nice cones"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4043","kind":"arxiv","version":4},"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":"sha256:91cc8a8e4f8fb8706a2a4797c90d5dbd786f8bc785972f09d1f42be3694fe15d","target":"record","created_at":"2026-05-18T03:41:01Z","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":"bd2ccaa1abfea10ffe0f29fdb8ab1be55739b256bca8680f063e01e2ec4c913a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-02-18T00:21:09Z","title_canon_sha256":"6f7c1c50a59c2e07262b4ced1171e937e5065220775afead6ff91df9a971364e"},"schema_version":"1.0","source":{"id":"1202.4043","kind":"arxiv","version":4}},"canonical_sha256":"1b540f914d723c018107cfbf086a32a31d5c016185e7a8531c35d01cbcb4688c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b540f914d723c018107cfbf086a32a31d5c016185e7a8531c35d01cbcb4688c","first_computed_at":"2026-05-18T03:41:01.251505Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:01.251505Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8c76dDTvY7li/MMHbuvU2clegwLe7f9SXPkp+mNVxmPoynlnM1/MdciDZs3HbSRwkFwS6cIucniYwG6q5Lm8Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:01.252227Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.4043","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91cc8a8e4f8fb8706a2a4797c90d5dbd786f8bc785972f09d1f42be3694fe15d","sha256:f2b7afa75bb33d816529698fb0881019c0ec02cead94385ffea4dde52d6d188d"],"state_sha256":"31f4a8b0c631a00bfd321be51c07e3eaaa955647b1d83bd31cfe8a8f356a8a93"}