The similarity between
triangulations of the sphere and simplicial polytopes makes cells with triangulated
boundaries natural generalizations of simplicial polytopes. In this paper we extend
this generalization to cells whose boundaries are broken up into more general
structures than just simplices. These structures are called gcc’s. In doing so we
get a generalization of the d-polytope. We shall investigate a method of
constructing these structures, called facet splitting. We show that almost all
d-gcc’s with up to 3 + d facets can be constructed by facet splitting, and we
construct a simple 4-gcc with 10 facets that cannot be constructed in this
way.
