  We consider the polytope arising from a marked surface by flips of
triangulations. Sleator, Tarjan and Thurston studied in 1988 the diameter of
the associahedron, which is the polytope arising from a marked disc by flips of
triangulations. They showed that every shortest path between two vertices in a
face does not leave that face. We establish that same non-leaving-face property
for all unpunctured marked surfaces.
