Exercise 20, § 5

Let A, B be two disjoint closed convex sets in a Hausdorff locally convex space E and let H be a closed hyperplane separating A from B; suppose that $A \cap H \neq \varnothing$ and that the intersection of $A \cap H$ and of every line is compact. Show that, if A or B is locally compact, then there exists a neighbourhood V of 0 in E such that $(A + V) \cap B$ is empty. (Consider two cases according to whether A or B is locally compact; in the first case, note that there exists a hyperplane $H'$ parallel to H such that, if S is the set of points between H and $H'$, then $A \cap S$ is compact. In the second case, suppose for example that $0 \in B \cap H$; for every neighbourhood V of 0 in E, consider the set $(A + V) \cap B$ and consider successively the case where this set is relatively compact for at least one V or the case when this is not so, as in exerc. 16 of II, p. 67.)