In Hales's book (cited below), he associates what he calls a *fan* with any convex polyhedron in $\mathbb{R}^3$.
I will not define his notion of fan, but let his figure (p.137) serve as a definition:

My question is: Has this natural object been defined and used previously in other
contexts and perhaps under another name?

Thomas Hales,

Dense Sphere Packings: A Blueprint for Formal Proofs. 2012. (Cambridge link)