This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help would be appreciated.

Definition: The book defines a slice of a subset $C$ of a Banach space $X$, to be a nonempty intersection with an open half space of $X$.

Lemma: Let $X$ be a Banach space, and let $x$ be an extreme point for its unit ball. Assume that the relative norm and the weak topology coincide at $x$. Show that the slices form a neighborhood base of the norm topology.