{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FLH5UU7O7XXTMGSTBKGSHWV2XE","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":"2bb909ab40533a4578d6bc2087013b1bc7dd274fb3131f7074885f4fcc28337b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-28T15:44:08Z","title_canon_sha256":"7a6042a8be370a6a90cc1d721f4edb97f53d4489f90707655430940a19de25ec"},"schema_version":"1.0","source":{"id":"1604.08473","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08473","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08473v4","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08473","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"pith_short_12","alias_value":"FLH5UU7O7XXT","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FLH5UU7O7XXTMGST","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FLH5UU7O","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:47bc065f4c3599b29af538acdef79866ae4ea187d1bc39568396118d17941d3b","target":"graph","created_at":"2026-05-18T01:11:17Z","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 Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $\\Phi$-convexity. Under general conditions on the class of functions $\\Phi$, the Krein-Milman-Ky Fan theorem asserts then, that every compact $\\Phi$-convex subset of a Hausdorff space, is the $\\Phi$-convex hull of its $\\Phi$-extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the ","authors_text":"Mohammed Bachir","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-28T15:44:08Z","title":"On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08473","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":"sha256:62d2bf46bedb8c89b795fe53b57b74f5d4fa9dab2257df061c60b9cd2a9b4213","target":"record","created_at":"2026-05-18T01:11:17Z","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":"2bb909ab40533a4578d6bc2087013b1bc7dd274fb3131f7074885f4fcc28337b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-28T15:44:08Z","title_canon_sha256":"7a6042a8be370a6a90cc1d721f4edb97f53d4489f90707655430940a19de25ec"},"schema_version":"1.0","source":{"id":"1604.08473","kind":"arxiv","version":4}},"canonical_sha256":"2acfda53eefdef361a530a8d23dabab90717d8ce646e336c32049755d389d566","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2acfda53eefdef361a530a8d23dabab90717d8ce646e336c32049755d389d566","first_computed_at":"2026-05-18T01:11:17.246142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:17.246142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w8XMlq7pUV9KClcFpZX85v2RAoDSUPwAvraWp9m1V56qkO8RUzsskBnQPi4XfpKfkOjoCEEuI6tpOZX+RjSTDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:17.246726Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.08473","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62d2bf46bedb8c89b795fe53b57b74f5d4fa9dab2257df061c60b9cd2a9b4213","sha256:47bc065f4c3599b29af538acdef79866ae4ea187d1bc39568396118d17941d3b"],"state_sha256":"d7d47912f750e3e9a15606dbc99368b3e68069a793976d136258cf476fb32302"}