Near classification of compact hyperbolic Coxeter d-polytopes with d+4 facets and related dimension bounds

We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial types of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 51 polytopes, respectively, yielding many new examples for further study (also discovered independently by Ma and Zheng).

We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $k\le 10$. It was shown by Vinberg in 1985 that for any $k$, we have $d\le 29$, and no better bounds have previously been published for $k\ge 5$. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.