Closed convex hull of a set is the unit ball.

Question

Let $X$ be a (real) Banach space and $A \subseteq X^{*}$ some weak*-compact subset of a unit ball in $X^*$. Furthermore, assume that for any functional $f \in A$, we have $-f \in A$. Now we know that, as a corollary of Hahn-Banach, for any $x \in X$ we have $$\| x\| = \sup \{|f(x)|\ |\ f \in X^*,\ \| f\| \leq 1\}.$$ Is it true that if
$$\| x\| = \sup \{|f(x)|\ |\ f \in A\},\quad \forall x \in X,$$ then the closed convex hull of $A$ is the unit ball of $X^*$? If yes, why? If not, under which conditions would this hold?