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.

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.

EDIT: The book advices to consider the point $j(x)$, where $j$ is the naturall isometry to the by-dual and prove that it is an extreme point for the unit ball there. To this end assume $j(x)$ is not extreme, i.e. $j(x)=\frac{1}{2}(x_1+x_2), x_i\in B_{X^{**}}$

Now by a geometric argument show that the relative $\omega^\ast$, and the relative norm topologies for $B_{X^{**}}$ coincide at $x_i$ and so forth.

p.s. I believe the statement in the last sentence is the essence of the question and of potential interest to this community. At least it wasn't getting any attention on Math.SE, hence I asked them to migrate it here. Any help would be appreciated.