The Associahedron and Triangulations of the n-gon

Let Pn be a convex n-gon in the plane, n ⩾ 3. Consider Σn, the collection of all sets of mutually non-crossing diagonals of Pn. Then Σn is a simplicial complex of dimension n − 4. We prove that Σn is isomorphic to the boundary complex of some (n − 3)-dimensional simplicial convex polytope, and that this polytope can be geometrically realized to have the dihedral group Dn as its group of symmetries. Formulas for the f-vector and h-vector of this polytope and some implications for related combinatorial problems are discussed