So in this post:
Faster Algorithms for Convex Hulls
I was interested in determining if a convex hull of two $n$ dimensional polyhedra can be computed quickly, and the answer was in general: no, since for certain pairs of polyhedra the number of facets in the hull grows exponentially with the number of facets of the original constituents.
Now a natural question that follows is, given two polyhedra is it possible to estimate how many facets will be on their convex hull? What about how many facets up to a multiplicative factor $\epsilon$ of accuracy?
Ideas:
I'm really lost as far as techniques go. As the post revealed, two very innocent looking n-simplexes if placed in the right orientation can send you to $2^n$ facets, yet in another orientation result in only $2n$ facets, both almost equally spaced. It seems that once you know the shape and orientation of the pair, understanding how they are located w.r.t each other plays a crucial rule in determining the facet count.