New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-space

Abstract

In this paper we consider ball packings in $4$-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $4$-dimensional realizable packing density upper bound due to L. Fejes T\'oth (Regular Figures, 1964). We give seven examples of horoball packing configurations that yield higher densities of $0.71644896\dots$, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.