Weak-Star Convergence of Convex Sets

We show that if a Banach space $X$ is weakly compactly generated and $C$, $C_n$ are weak-star-closed bounded convex nonempty subsets of the dual space $X^*$, then the support functionals $\delta^*_{C_n}$ converge to $\delta^*_C$ pointwise on $X$ if and only if the sequence $(C_n)$ is uniformly bounded with weak-star limit $C$.