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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 sameface $f$, then any geodesic between $v$ and $v'$ does not leave $f$. In type$A$, we show that our geometric projection recovers a slight modification ofthe combinatorial projection given by D. Sleator, R. Tarjan, and W. Thurston.

W-Associahedra are In-Your-Face