Secondary fans and secondary polyhedra of punctured Riemann surfaces

A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration $A \subset \mathbb{R}^d$ a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of $A$. That fan arises as the normal fan of a convex polytope. In a completely analogous way we associate to each hyperbolic Riemann surface $R$ with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of $R$ that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of $R$ turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of $R$.