Exercise 13, § 5

Let A and B be two closed convex sets in the space $\mathbf{R}^n$, such that if V and W are affine linear varieties generated by A and B respectively, then no point of $A \cap B$ is both interior to A relative to V and interior to B relative to W. Show that there exists a hyperplane separating A from B. (By taking quotients, reduce it to the case where either one of the varieties V, W is contained in the other or V and W are complementary vector subspaces in E.)