Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be different, of course. But is this information enough to reconstruct number of faces of $P$ of all dimensions? If yes, what is specific procedure to do this? I am looking at what happened when we move facets parallel and pass through `singularity', i.e. not-simple polytope, and on first glance it looks that $f$-vector preserves, but I am always not sure with such things. And this is not good proof anyway, I would prefer more direct argument.

If the statement is however false, I wonder for which specific collections it is still true.

Every simple polytope with the same set of facet normals has the same $f$-vector. $\endgroup$ – Joseph O'Rourke Oct 17 '15 at 19:03