This is related to one that I asked earlier:

The intersection of two $l_1$ balls

My question: are there any general classes of polytopes $P_1,P_2\subset\mathbb{R}^n$ such that $P_1$ has few vertices, $P_2$ has few vertices, and $P_1\cap P_2$ has few vertices? (in my particular problem, $P_1$ is a unit ball in the $l_1$ norm, but I am interested in this question more generally)