W-Associahedra are In-Your-Face

We use a projection argument to uniformly prove that $W$-permutahedra and $W$-associahedra have the property that if $v,v^{\mathrm{\prime }}$ are two vertices on the same face $f$, then any geodesic between $v$ and $v^{\mathrm{\prime }}$ does not leave $f$. In type $A$, we show that our geometric projection recovers a slight modification of the combinatorial projection given by D. Sleator, R. Tarjan, and W. Thurston.