This is related to one that I asked earlier:

http://mathoverflow.net/questions/201762/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, but I am interested in this question more generally)