Polytopal realizations of generalized associahedra
  In hep-th/0111053, a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a generalized associahedron of the corresponding type). Here we prove this conjecture by explicitly exhibiting a family of such polytopal realizations. 
Frederic Chapoton, Sergey Fomin, and Andrei Zelevinsky