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?



Conditions for differentiability of minima and minimizers of linear functionals?
In general $V$ is Lipschitz, but it is not smooth: take e.g. $B=\mathbb{R^2}$ with the Euclidean norm, and let $C$ be a segment, say $C:=[-1,1]\times(0)$. Then, for all $(x,y)\in B^\*=B$ we have $$V(x,y)=\min\_{|c|\le1} \ cx = \min \{x,-x\}\, .$$ To get a smooth $V$, smoothness conditions on $C$ are required.


