Thanks but that is not what I meant - given an extreme point of the dual ball, can it be expressed as a norm limit of extreme points which norm something? No convex hull taken.

As an aside, is there then an analogue of the Bishop-Phelps theorem restricted to extreme norm one linear functionals? All I can prove is weak$^*$ density (which is enough for my needs).

To explain II, maybe a quick proof is best: You can reduce fairly easily to the case where $X$ is the indicator function of some event $A$. Let $f = P[A|{\cal F}]$. Then the inequality becomes $E[f^2] \geq E[f]^2$, i.e., $V(f) \geq 0$, and equality holds iff $V(f) = 0$, i.e., if $f$ is constantly $E[f] = P[A]$, i.e., if $A$ is independent from $\cal F$.

Thank you, but I do know how to prove these things - my question is whether this exists in the literature.