Exercise 27, § 7

Let $C$ be a closed convex set in a product of lines, $E$, and let $a$ be an extremal point of $C$. Show that for every neighbourhood $V$ of $a$ in $C$, there exists an open half-space $F$ in $E$ such that $a \in F \cap C \subset V$. (Reduce to the case when $C$ is compact.)