Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$. For every continuous linear functional $F$ on $B$, define $V(F)=min_{c\epsilon C} F(c)$ and $S(F)= { \lbrace c \epsilon C : V(c)=V(F)\rbrace }$.

Is it true (or are there known conditions under which) $V:B^*\rightarrow \mathbb{R}$ is Frechet differentiable?

When $S(F)$ is known to be single valued and hence can be seen as a function $S:B^* \rightarrow B$, are there conditions under which it is known to be Frechet differentiable?

When $S(F)$ is a set valued map, is there some other sense in which it might be differentiable?