{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QZ47DWIF34OARFIKGIDOX367PX","short_pith_number":"pith:QZ47DWIF","canonical_record":{"source":{"id":"1803.06573","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2018-03-17T21:29:20Z","cross_cats_sorted":[],"title_canon_sha256":"27ecda032ec7dc454997be317c78cc7c50465582ef0abfd0731a51bd7360aff0","abstract_canon_sha256":"27a11aad31bcc35e5c9138a1bba97b5c1a78c4d5c0034344867dd766dc57d571"},"schema_version":"1.0"},"canonical_sha256":"8679f1d905df1c08950a3206ebefdf7dd6dc4d67c665d5926b55bde24a5710cc","source":{"kind":"arxiv","id":"1803.06573","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06573","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06573v1","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06573","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"pith_short_12","alias_value":"QZ47DWIF34OA","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"QZ47DWIF34OARFIK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"QZ47DWIF","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QZ47DWIF34OARFIKGIDOX367PX","target":"record","payload":{"canonical_record":{"source":{"id":"1803.06573","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2018-03-17T21:29:20Z","cross_cats_sorted":[],"title_canon_sha256":"27ecda032ec7dc454997be317c78cc7c50465582ef0abfd0731a51bd7360aff0","abstract_canon_sha256":"27a11aad31bcc35e5c9138a1bba97b5c1a78c4d5c0034344867dd766dc57d571"},"schema_version":"1.0"},"canonical_sha256":"8679f1d905df1c08950a3206ebefdf7dd6dc4d67c665d5926b55bde24a5710cc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:45.570464Z","signature_b64":"RCP0jre5CoC7LSqKhAojWY2tVUFFJGqavCMTsXsdoY2tmNcuWO44zUsSb/2+6ZyNQAPo61YsK/7WarNNYPNiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8679f1d905df1c08950a3206ebefdf7dd6dc4d67c665d5926b55bde24a5710cc","last_reissued_at":"2026-05-18T00:20:45.569944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:45.569944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.06573","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":"2026-05-18T00:20:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VZx8apHE6bNnqM3L9IAQjt4WrFKakuFE/xIRgvbhRan6LRmbTkR8wkD8pWVGzxgumbt/WorwOkf9diOqu+pZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:47:35.733108Z"},"content_sha256":"752ecce57cae8862dd38f0eb4031a5c55459c6bc3f628c2e0a327dbf74094ea5","schema_version":"1.0","event_id":"sha256:752ecce57cae8862dd38f0eb4031a5c55459c6bc3f628c2e0a327dbf74094ea5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QZ47DWIF34OARFIKGIDOX367PX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Xingyu Zhou","submitted_at":"2018-03-17T21:29:20Z","abstract_excerpt":"We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivale"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06573","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":"2026-05-18T00:20:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uFa9dbXc8rLWOzd5Cfbi8bYGccU5XfvTYFO/bZItHwHHi9IPiWEKtQ5/P7rqSGAk5+v/aw//CCGPKwpEEDYmDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:47:35.733797Z"},"content_sha256":"02eb9fb77d37a20e48a3fadc8b2a05771d9ce0dbf685c57bbceb36e9224c006d","schema_version":"1.0","event_id":"sha256:02eb9fb77d37a20e48a3fadc8b2a05771d9ce0dbf685c57bbceb36e9224c006d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QZ47DWIF34OARFIKGIDOX367PX/bundle.json","state_url":"https://pith.science/pith/QZ47DWIF34OARFIKGIDOX367PX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QZ47DWIF34OARFIKGIDOX367PX/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":"2026-05-25T22:47:35Z","links":{"resolver":"https://pith.science/pith/QZ47DWIF34OARFIKGIDOX367PX","bundle":"https://pith.science/pith/QZ47DWIF34OARFIKGIDOX367PX/bundle.json","state":"https://pith.science/pith/QZ47DWIF34OARFIKGIDOX367PX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QZ47DWIF34OARFIKGIDOX367PX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QZ47DWIF34OARFIKGIDOX367PX","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":"27a11aad31bcc35e5c9138a1bba97b5c1a78c4d5c0034344867dd766dc57d571","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2018-03-17T21:29:20Z","title_canon_sha256":"27ecda032ec7dc454997be317c78cc7c50465582ef0abfd0731a51bd7360aff0"},"schema_version":"1.0","source":{"id":"1803.06573","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06573","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06573v1","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06573","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"pith_short_12","alias_value":"QZ47DWIF34OA","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"QZ47DWIF34OARFIK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"QZ47DWIF","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:02eb9fb77d37a20e48a3fadc8b2a05771d9ce0dbf685c57bbceb36e9224c006d","target":"graph","created_at":"2026-05-18T00:20:45Z","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":"We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivale","authors_text":"Xingyu Zhou","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2018-03-17T21:29:20Z","title":"On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06573","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":"sha256:752ecce57cae8862dd38f0eb4031a5c55459c6bc3f628c2e0a327dbf74094ea5","target":"record","created_at":"2026-05-18T00:20:45Z","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":"27a11aad31bcc35e5c9138a1bba97b5c1a78c4d5c0034344867dd766dc57d571","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2018-03-17T21:29:20Z","title_canon_sha256":"27ecda032ec7dc454997be317c78cc7c50465582ef0abfd0731a51bd7360aff0"},"schema_version":"1.0","source":{"id":"1803.06573","kind":"arxiv","version":1}},"canonical_sha256":"8679f1d905df1c08950a3206ebefdf7dd6dc4d67c665d5926b55bde24a5710cc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8679f1d905df1c08950a3206ebefdf7dd6dc4d67c665d5926b55bde24a5710cc","first_computed_at":"2026-05-18T00:20:45.569944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:45.569944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RCP0jre5CoC7LSqKhAojWY2tVUFFJGqavCMTsXsdoY2tmNcuWO44zUsSb/2+6ZyNQAPo61YsK/7WarNNYPNiCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:45.570464Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.06573","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:752ecce57cae8862dd38f0eb4031a5c55459c6bc3f628c2e0a327dbf74094ea5","sha256:02eb9fb77d37a20e48a3fadc8b2a05771d9ce0dbf685c57bbceb36e9224c006d"],"state_sha256":"d00d83623e09c0b42859188c185e8e97dcaae4c9a11702eedbfbc119ba4d5c8d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"toXnti+UnxTn/6d6joogIDwy5ybaMWEfQ99S8ix3ihDFDukb+IklIhz/7H/Dwdp7lvsjvCfFsnleqzNdQIvhAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T22:47:35.737129Z","bundle_sha256":"f0a3047cfd5422255025f4eaf140c10a9098291146346bea0e027881ab4db9e7"}}