Isometry classes of generalized associahedra

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclideanspacewithsimpleroots$\Delta={\a_s | s\in S}$ of the same length and fundamental weights ${\mathrm{\Delta }}^{\ast }=\{v_s\mathrm{\mid }s\in S\}$. We set $M(e)={\sum }_{s\in S}{\kappa }_s{v}_s$, ${\kappa }_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w)\mathrm{\mid }w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Ass{o}_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Ass{o}_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c^{\mathrm{\prime }}$ two Coxeter elements in $W$, we have that $Ass{o}_c(W)$ and $Ass{o}_{c^{\mathrm{\prime }}}(W)$ are isometric if and only if $\mu (c^{\mathrm{\prime }})=c$ or $\mu (c^{\mathrm{\prime }})=w_0{c}^{-1}{w}_0$ for $\mu$ an automorphism of the Coxeter graph of $W$ such that ${\kappa }_s={\kappa }_{\mu (s)}$ for all $s\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.