The following thing are known about hyperbolic right-angled polytopes:
- Compact hyperbolic right-angled polytopes do not exist in dimensions greater than 4 (Vinberg, Potyagailo). Examples are known up to dimension 4 (folklore?).
- Hyperbolic right-angled polytopes of finite volume do not exist in dimensions greater than 12 (Dufour). Examples are known up to dimension 8 (Vinberg, Potyagailo).
- Ideal (i.e. all vertices are ideal) right-angled polytopes do not exist in dimensions greater than 6 (Sasha). Examples are known up to dimension 4 (folklore?).
All the results 1-2-3 use the Nikulin-Khovanskii inequality, bounding the average number $f^k_l$ of $k$-dimensional faces in $l$-dimensional ones for an $n$-dimensional convex polytope (not necessarily hyperbolic). Here $k,l \leq [n/2]$ is a condition.
The bound has the form $f^k_l < g(k,l,n)$, where $g$ is a decreasing function in $n$, if $k$ and $l$ are fixed. Thus, by getting a lower bound for $f^k_l$, we can bound $n$ from above.
The result by Vinberg Potyagailo uses the fact that $5 \leq f^1_2$, for a compact right-angled polytope, that it obtains the best possible bound on $n$ (w.r.t. the given $k,l$). The result by Dufour uses the fact that $27 \leq f^5_6$ for a finite-volume right-angled polytope. Thus, the best possible bound for $n$ with given $k,l$. Finally, I know that $24 \leq f^3_4$ for an ideal right-angled polytope and get the best possible bound with given $k,l$. Again.
It's clear that increasing $k,l$ will not improve the bound on $n$ in any of the above cases 1-2-3. However, decreasing $k,l$ (and finding the respective lower bounds for $f^k_l$ in each of the cases 1-2-3) does not improve it either. Thus, the only sharp bound coming from the use of the Nikulin-Khovanskii inequality happens in case 1.
My question is the following: Can one find a way of proving sharp bounds in cases 2-3 (let me have a liberty to conjecture that the respective bounds are 8 and 4) without making use of the Nikulin-Khovanskii inequality? Basically, I'm interested in any fact that helps settling the conjectural bounds, just at the moment I see that the Nikulin-Khovanskii inequality is not (directly) applicable any more.