  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 given by D. Sleator, R. Tarjan, and W. Thurston.
