This is a known result that I see everywhere yet I didn't manage to find a proof or a reference.
Let $X$ be a compact convex subset of a locally convex TVS $E$, then $X=\overline{\mathrm{conv}} ~\mathrm{ext}(X)$.
I wonder if there is a simple proof of that, I also wonder if this implies that if $X$ is closed and bounded (and convex) then $X=\overline{\mathrm{conv}} ~\mathrm{ext}(X)$, indeed $X$ is compact in the weak topology. Any reference is most welcome.
Closed convex and bounded sets need not to be a closed convex envelope of the set of its extreme points even in Banach space. So, for example in the Banach space $c_0 $ the unit closed ball is closed convex and bounded set but has not posses any extreme point hence it can not be a closed convex envelope of the set of its extreme points.
But the result is true in reflexive Banach spaces since in reflexive spaces the weak topology and the weak* topology coincide.