If $A$ and $B$ are convex and disjoint then there exist a continuous linear function such that $f(A)$ and $f(B)$ are dijoint.

Question

This looks similar to the hyperplane Hahn-Banach separation theorem except that we may have empty interior. I want to know if the following is true.

Let $X$ be a real locally convex topological vector space and let $A$ and $B$ be convex disjoint subsets, then there exists a continuous linear functional $f$ on $X$ such that $f(A)\cap f(B)$ are disjoint.

I would be interested in any approach to this problem or in any counter example.