Diameters and geodesic properties of generalizations of the associahedron

The n-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex (n + 3)-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is 2n−4 as soon as n > 9. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both.