Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define its support function $s(.|C):X^*\to \overline{\mathbb{R}}$ by: $$ s(x^*|C):=\sup_{x\in C} \langle x^*,x \rangle\qquad \forall x^*\in X^* $$ and its radius by :
$$ \|C\|:=\sup_{x\in C}\|x\| $$
then : $$ s(x^*|C)\le \|x^*\|\|C\|\qquad \forall x^*\in X^* $$
We define the hausdorff distance $h$ on $\mathcal{P}_{wkc}(X)$ by : let $A,B\in \mathcal{P}_{wkc}$ : $$ h(A, B) = \sup_{\|x^*\|\leq 1} |s(x^*|A)-s(x^*|B)| $$ We shall denote by $\tau$ the Mackey topology on the dual space $X^*$ such that :
$$ x_n^*\overset{\tau}{\longrightarrow} x^* \Longleftrightarrow \forall C\in \mathcal{P}_{wkc}~:~\sup_{x\in C}{\langle x_n^*-x^*,x\rangle}\to 0 $$
Let $D\subset X^*$ be a subset $\tau$-dense in unit ball of $X^*$ $\big(\{x^*\in X^*~:~\|x^* \|\le 1\}\big)$.
Show that : $$ h(A, B) = \sup_{x^*\in D} |s(x^*|A)-s(x^*|B)| $$ An idea please.