For a convex polytope $P$ in $\mathbb R^4$, denote by $N\_0,N\_1,N\_2,N\_3$ respectively the number of vertices, edges, faces, cells. By Euler's formula, we know $N\_0+N\_2=N\_1+N\_3$, which means there is a sort of equilibrum among the $N\_i$. But I wonder if there exist upper and/or lower bounds for $f(P):=\frac{N\_1+N\_2}{N\_0+N\_3}$ where $P\subset\mathbb R^4$ is any convex polytope.


E.g. for the regular one called 24-cell, we have $f(P)=4$.



convex polytopes with many faces and edges but few cells and vertices 
As of 2003, I don't think it was known whether $f(P)$ is bounded. See *Fat 4-polytopes and fatter 3-spheres* by Eppstein, Kuperberg, and Ziegler (Monogr. Textbooks Pure Appl. Math. **253** (2003), 239-265, doi:[10.1201/9780203911211](https://doi.org/10.1201/9780203911211), arxiv:[math/0204007](https://arxiv.org/abs/math/0204007)), which contains a beautiful construction that achieves $f(P)>5.048$, and shows that it is not bounded for strongly regular CW decompositions of the 3-sphere (instead of polytopes). I don't know whether there has been more recent progress.


