Conic James' Compactness Theorem

The following results is proved: Let $A$ be a convex bounded non weakly relatively compact subset of a Banach space $E$. We consider a convex weakly compact subset $D$ of $E$ which does not contain the origin. Then there is a sequence $\left\{x_n^*\right\}_{n\ge 1}$ in $B_{E^*}$ and $g_0^*\in \hbox{co}_{\sigma}\{x_n^*:n\ge 1\}$ such that for all $h\in \ell_\infty (A)$ satisfying that for all $a\in A,$ $$\liminf_{n\ge 1}x_n^*(a) \le h(a) \le\limsup_{n\ge 1}x_n^*(a),$$ we have that $g_0^*- h$ does not attain its supremum on $A$ and $( g_0^*- h)(d)>0$ for every $d\in D$.