Is it true that given a real vector space $X$ and two disjoint convex sets $A,B\subseteq X$, there is always a linear functional that (weakly) separates them? I.e., is there a non-zero linear functional $\phi\colon X\to \mathbb R$ and a $\gamma\in\mathbb R$, such that $\phi(a)\le \gamma \le \phi (b)$ for all $a\in A$ and $b\in B$? If not, can you give a counterexample?
I know the statement is true if you add the aditional hypothesis that at least one of the sets has an internal point. This follows from the Hahn-Banach theorem using the Minkowski functional, but I was wandering whether this hypothesis is necessary.