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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.

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    $\begingroup$ Any own work on this? -- It would have been nice to make sure that you copy the hint correctly... 1. It must be the relative weak*-topology of $B_{X^{\ast\ast}}$ (not the weak topology) at $x_i$ that coincides with the relative norm topology. 2. There is no reason for the $x_i$ to be extremal points of $B_{X^{**}}$. 3. An explanation of the word "slices" might help for those who don't have the book, and, finally, 4. the name is Goldstine, not Goldstein. $\endgroup$
    – commenter
    Jun 5, 2013 at 11:50
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    $\begingroup$ I believe the paper you seek is $$ $$ MR0928972 (89e:46016) Reviewed Lin, Bor-Luh(1-IA); Lin, Pei-Kee(1-MEMP); Troyanski, S. L.(BG-SOFI) Characterizations of denting points. Proc. Amer. Math. Soc. 102 (1988), no. 3, 526–528. 46B20 (52A07) $\endgroup$
    – Bill Johnson
    Sep 27, 2013 at 19:30

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