  In this paper we study the horoball packings related to the hyperbolic 24
cell in the extended hyperbolic space $\overline{\mathbf{H}}^4$ where we allow
{\it horoballs in different types} centered at the various vertices of the 24
cell.
  We determine, introducing the notion of the generalized polyhedral density
function, the locally densest horoball packing arrangement and its density with
respect to the above regular tiling. The maximal density is $\approx 0.71645$
which is equal to the known greatest ball packing density in hyperbolic 4-space
given in \cite{KSz14}.
