Hahn-Banach theorems establish in a normed space several situations in which one can separate certain kinds of disjoint convex sets, but none of them comes without requirement -- such as one of the two convex sets must be open, or one is compact and the other one closed. Now let's consider different: we now put a much more stringent requirement on the normed space -- we require them to be real Euclidean spaces, but now let's throw away all the hypotheses about the convex sets except disjointness, that is, we only require the two convex sets to be disjoint. The point is, how to prove there exists at least one hyperplane separating the two sets? More formally,
Let $X=\Bbb R^n$, $A,B\subset X$ are disjoint convex sets, then there exist $v\in X$ and $c\in\Bbb R$ such that $v^Ta\le c\le v^Tb,\quad \forall a\in A,b\in B.$
Would you please give me any hint for a proof?