W-associahedra have the non-leaving-face property

We use a projection argument to uniformly prove that W-permutahedra and W-associahedra have the property that if v,v′ are two vertices on the same face f, then any geodesic between v and v′ does not leave f. In type A, we show that our geometric projection recovers a slight modification of the combinatorial projection in Sleator et al. (1988).