W-Associahedra are In-Your-Face

Abstract

We use a projection argument to uniformly prove that $W$-permutahedra and $W$-associahedra have the property that if $v,v^{\prime }$ are two vertices on the same face $f$, then any geodesic between $v$ and $v^{\prime }$ 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.

14 pages, 7 figures. Added a section relating the geometric and combinatorial projections in type A

Cited by in corpus (1)

• Non-Leaving-Face property for marked surfaces