Polyhedra intersect subspace

Hi,

I am having trouble with a certain geometric question and would be grateful if someone can give pointers. The question is as follows:

I have a set of polyhedra in n-dimensional space, and I want the smallest dimensional subspace that intersects with all of the polyhedra.

If it helps, assume the polyhedra are all cones pointed at 0, and the subspace should not be the `0' subspace.

P.S: I have the cdd program by Komei Fukuda, and hence can switch between the two representations of polyhedra (linear inequalities or extremal points).

Thanks,

Harish.