Conditions for differentiability of minima and minimizers of linear functionals?

Question

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?

Answer 1

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.