I am having trouble proving the following property. Maybe the result already exists, but I could not find anything on the topic.
Proposition: Let $P_1$ and $P_2$ be two disjoint polytopes of $\mathbb{R}^n$ (for instance the convex hulls of two finite sets of points). The smallest affine convex cone $\mathcal{C}$ ("smallest" relatively to the whole $\mathbb{R}^n$ space) such that $P_1 \in \mathcal{C}$ and $P_2 \in -\mathcal{C}$ has extreme rays defined by some couples of vertices $v_j-v_i$ where $v_j$ is an extreme point of $P_2$ and $v_i$ an extreme point of $P_1$.
The provided figure gives an example (consider the red polytopes) in $\mathbb{R}^2$, the result in that case seems quite natural : http://imageshack.com/a/img910/1357/H4DcOu.png. On it, one can observe that there indeed exists an affine cone (it is pointed on a point that is not the origin) than contains one polytope while its opposite cone contains the other polytope.
I do not know how to prove it in the generic case of $\mathbb{R}^n$. I guess a geometric proof would be reachable in $\mathbb{R}^2$ but : I am afraid to be missing some more obvious/elegant proof, and I don't know if this could be transposable to $\mathbb{R}^n$.
Thanks in advance !