Exercise 19, § 5

Let C, $C'$ be two closed convex sets in a Hausdorff topological vector space E that are strictly separated by a closed hyperplane H. Let $H'$ be a closed hyperplane of support to both C and $C'$ such that C and $C'$ lie on the same side of $H'$. Show that $H'$ is the only hyperplane with these properties which contains $H \cap H'$ and that $H \cap H'$ is a support hyperplane of the trace P on H of the convex envelope of $C \cup C'$. Conversely, if C and $C'$ are compact, then for every support hyperplane D of P in H, there exists a hyperplane $H'$ that supports both C and $C'$, which contains D and such that C and $C'$ lie on the same side of $H'$.