{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:YCYL2QXTYLKV55Q5YOGCOSCSV5","short_pith_number":"pith:YCYL2QXT","canonical_record":{"source":{"id":"1804.10738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-04-28T03:57:45Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"a0653cc324d09ac5afe71711157e36d122686298a3f71146e5abfbceb64092c9","abstract_canon_sha256":"e6daf4b26e8f8ec304ec8c2535f555896b2cd07fc5ed824ec4d1b995df7039a3"},"schema_version":"1.0"},"canonical_sha256":"c0b0bd42f3c2d55ef61dc38c274852af670078f0430d82c958f8524b562a9109","source":{"kind":"arxiv","id":"1804.10738","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10738","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10738v2","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10738","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_12","alias_value":"YCYL2QXTYLKV","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_16","alias_value":"YCYL2QXTYLKV55Q5","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_8","alias_value":"YCYL2QXT","created_at":"2026-07-05T00:09:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:YCYL2QXTYLKV55Q5YOGCOSCSV5","target":"record","payload":{"canonical_record":{"source":{"id":"1804.10738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-04-28T03:57:45Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"a0653cc324d09ac5afe71711157e36d122686298a3f71146e5abfbceb64092c9","abstract_canon_sha256":"e6daf4b26e8f8ec304ec8c2535f555896b2cd07fc5ed824ec4d1b995df7039a3"},"schema_version":"1.0"},"canonical_sha256":"c0b0bd42f3c2d55ef61dc38c274852af670078f0430d82c958f8524b562a9109","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:09:12.561867Z","signature_b64":"igg40XgeVik8x8ZOGDE1jmeIXDTX5KOk8RZu+e5XLBa8j4c6DDz07VucwI3I6+tm6+7pLBQFLl3B9YGQ75cSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0b0bd42f3c2d55ef61dc38c274852af670078f0430d82c958f8524b562a9109","last_reissued_at":"2026-07-05T00:09:12.561448Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:09:12.561448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.10738","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":"2026-07-05T00:09:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dxY4RID4us8KiJXeYfhPCN5wcrdKzTRTIIQvqWVFrPTWoCPzuFdbs8uqfEzUkZg0/mRzSeXV483XjlclwuXpAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:05:44.317043Z"},"content_sha256":"2cab0fa82e7c8bbf64bbd8813af02b7cbd054901e4b506d37c5fa99b4307314f","schema_version":"1.0","event_id":"sha256:2cab0fa82e7c8bbf64bbd8813af02b7cbd054901e4b506d37c5fa99b4307314f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:YCYL2QXTYLKV55Q5YOGCOSCSV5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Riemannian Corollary of Helly's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.MG","authors_text":"Alexander Rusciano","submitted_at":"2018-04-28T03:57:45Z","abstract_excerpt":"We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr\\\"unbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least $\\frac{1}{n+1}$ of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10738","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/1804.10738/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":"2026-07-05T00:09:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eQeIs+jKNf3BOdGH5KIzzqwrnf22siRcCmJN1aca8T1yDSNfHKbIHbV5qvG+v6kk/LMZZmcg4H2PgOiUHYYIDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:05:44.317543Z"},"content_sha256":"272e2af458df3f2263848375d350588554014c4ceb03af93086508a5546f9d17","schema_version":"1.0","event_id":"sha256:272e2af458df3f2263848375d350588554014c4ceb03af93086508a5546f9d17"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/bundle.json","state_url":"https://pith.science/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/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":"2026-08-08T04:05:44Z","links":{"resolver":"https://pith.science/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5","bundle":"https://pith.science/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/bundle.json","state":"https://pith.science/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YCYL2QXTYLKV55Q5YOGCOSCSV5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YCYL2QXTYLKV55Q5YOGCOSCSV5","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":"e6daf4b26e8f8ec304ec8c2535f555896b2cd07fc5ed824ec4d1b995df7039a3","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-04-28T03:57:45Z","title_canon_sha256":"a0653cc324d09ac5afe71711157e36d122686298a3f71146e5abfbceb64092c9"},"schema_version":"1.0","source":{"id":"1804.10738","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10738","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10738v2","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10738","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_12","alias_value":"YCYL2QXTYLKV","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_16","alias_value":"YCYL2QXTYLKV55Q5","created_at":"2026-07-05T00:09:12Z"},{"alias_kind":"pith_short_8","alias_value":"YCYL2QXT","created_at":"2026-07-05T00:09:12Z"}],"graph_snapshots":[{"event_id":"sha256:272e2af458df3f2263848375d350588554014c4ceb03af93086508a5546f9d17","target":"graph","created_at":"2026-07-05T00:09:12Z","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/1804.10738/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr\\\"unbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least $\\frac{1}{n+1}$ of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem.","authors_text":"Alexander Rusciano","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-04-28T03:57:45Z","title":"A Riemannian Corollary of Helly's Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10738","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":"sha256:2cab0fa82e7c8bbf64bbd8813af02b7cbd054901e4b506d37c5fa99b4307314f","target":"record","created_at":"2026-07-05T00:09:12Z","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":"e6daf4b26e8f8ec304ec8c2535f555896b2cd07fc5ed824ec4d1b995df7039a3","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-04-28T03:57:45Z","title_canon_sha256":"a0653cc324d09ac5afe71711157e36d122686298a3f71146e5abfbceb64092c9"},"schema_version":"1.0","source":{"id":"1804.10738","kind":"arxiv","version":2}},"canonical_sha256":"c0b0bd42f3c2d55ef61dc38c274852af670078f0430d82c958f8524b562a9109","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0b0bd42f3c2d55ef61dc38c274852af670078f0430d82c958f8524b562a9109","first_computed_at":"2026-07-05T00:09:12.561448Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:09:12.561448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"igg40XgeVik8x8ZOGDE1jmeIXDTX5KOk8RZu+e5XLBa8j4c6DDz07VucwI3I6+tm6+7pLBQFLl3B9YGQ75cSAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:09:12.561867Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10738","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2cab0fa82e7c8bbf64bbd8813af02b7cbd054901e4b506d37c5fa99b4307314f","sha256:272e2af458df3f2263848375d350588554014c4ceb03af93086508a5546f9d17"],"state_sha256":"858254193214aa1f75309a55a77f98fe7e75b5e72ac5db54cdded9b6c097e0fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eDzzPY/x9ZwHFnECgnpJDAnG45UCF0q3LrUssXIS5dsDXjDmiSvwO3vMiTmVnXFAPlc+Upk9nROA/WcEMvo+DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T04:05:44.322389Z","bundle_sha256":"8d69c85e9f7e3df410112a25402c4a9960b8941db32db0575ddce3dc007ee597"}}