Exercise 3, § 5

Let $A$ be a convex set in a topological vector space $E$ and $x_0$ be a point of $E$. Then, there exists a closed hyperplane $H$, containing $x_0$, and such that all the points of $A$ lie on the same side of $H$ if and only if there exists a non-pointed convex cone $C$ with vertex $x_0$, which contains at least one interior point and does not meet $A$. (For an example of a convex set $A \neq E$ which is not contained in any half-space defined by a hyperplane, see II, p. 65, exerc. 5.)