An Application of the Generalised James' Weak Compactness Theorem

We provide a short proof of following theorem, due to Delbaen and Orihuela and independently, P\'erez-Aros and Thibault. Let $A$ be a nonempty closed and bounded convex subset of a Banach space $(X,\|\cdot\|)$ and let $W$ be a nonempty weakly compact subset of $(X, \|\cdot\|)$. If we have \par\vskip2mm \centerline{$x_0^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0\}\ \ \ \text{and}\ \ \ \mathrm{argmax}(y^*|_A) \not= \varnothing$} \par\vskip2mm for each $y^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0$ and $\sup_{w \in W} |(x^*-x^*_0)(w)|<1\}$, then $A$ is weakly compact.