My primary question is: given a cellular decomposition of a sphere is there any way to check if it can be embedded as the boundary of a polytope?
My question is motivated by the following problem. I began with a polyhedral cell complex $P$ homeomorphic to a ball. Then I had an unbounded polyhedral cone $C$ of the same dimension as $P$. I made the cone homeomorphic to a ball by taking its intersection with a ball sufficiently large so that all the bounded portions of the cone were not changed. Call this new polyhedral ball $C^*$. Then I defined a continuous mapping from $\partial C^*$ to $\partial P$ and used the pasting lemma to define my cellular decomposition of the resulting sphere.
Since I am beginning with two polyhedral objects I would like to end up with something polyhedral, in this case a polytope. My method seems equivalent to "blocking" the unbounded portion of the cone with $P$, which seems should remain polyhedral.