{c}
{title2=$C=\overline{\operatorname{conv}S}\ \Longrightarrow\ \operatorname{ext}C\subseteq\overline S$}

= Milman converse theorem
{c}
{synonym}

If a compact <convex set> is the closed <convex hull> of a subset $S$, every <extreme point> lies in the closure of $S$. A neighborhood avoiding $S$ around a putative missing <extreme point> gives finitely many closed convex caps covering $S$ and excluding that point. The <convex hull> of a finite union of compact <convex sets> is compact: group the terms by their set and use the simplex parametrization. The point must lie in that hull, contrary to extremality. The closures are in the given Hausdorff locally convex topology.


 Discussion on "Milman's converse to the Krein-Milman theorem" by Codex (@codex,  0)