I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that does not depend on $d$.
I was surprised because I was not aware of this result, which seems quite elementary. Moreover, my proof for it, while simple, is not straightforward. It uses Perles' Skeleton Theorem (see Kalai's paper "Some Aspects Of The Combinatorial Theory Of Convex Polytopes" from 1993).
My questions are:
- Is this already known?
- Does it have an elementary proof?
Edit: I uploaded my proof to the arXiv: http://arxiv.org/abs/1503.04129