Minimal compact face argument
= Minimal compact face argument
A nonempty compact <face of a convex set> in a Hausdorff <locally convex space> contains an <extreme point>. Chains of compact faces have nonempty compact intersection. A minimal face supplied by the <Zorn lemma> must be a singleton, since a separating <linear functional> would otherwise expose a proper maximizer face. This is the existence step in the <Krein-Milman theorem>.