{"paper":{"title":"On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohammed Bachir","submitted_at":"2016-04-28T15:44:08Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08473","kind":"arxiv","version":4},"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"}