Write a linear optimization problem to find a hyperplane that strictly separates two disjoint polyhedra. 
Let $P_{1}=\left\{x\:|\: Ax\leq b\right\}$ and $P_{2}=\left\{x\:|\: Cx\leq d\right\}$ be two disjoint polyhedra. Write a linear optimization problem to find a hyperplane that strictly separates $P_{1}$ of $P_{2}$.


I prefer short, elegant solutions. Consider the hyperplane parameterized by $v \in \mathbb{R}^n$ and $w\in \mathbb{R}$. The problem of finding the maximally seperating hyperplane can be formulated as elegantly as
$$\max\{ t : v^T x \geq 1+t \; \forall x :Ax\leq b, \; v^T x  \leq 1-t \; \forall x:Cx\leq d\}.$$
Note that $w$ is fixed at $1$. With standard techniques from Robust Optimization, this can be reformulated as:
$$\max\{ t : -b^Ty \geq 1+t, \; A^Ty + v = 0, \; d^T z \leq 1-t, \; C^Tz = v, \; y \geq 0, z \geq 0 \}.$$
