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