Isometry classes of generalized associahedra

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Let (W,S) be a finite Coxeter system acting by reflections on an R-Euclidean space with simple roots Δ={ₛ | s∈ S} of the same length and fundamental weights Δ*={vₛ | s∈ S}. We set M(e)=∑ₛ∈ ₛκₛ vₛ, κₛ>0, and for w∈ W we set M(w)=w(M(e)). The permutahedron Perm(W) is the convex hull of the set {M(w) | w∈ W}. Given a Coxeter element c∈ W, we have defined in a previous work a generalized associahedron Assoc(W) whose normal fan is the corresponding c-Cambrian fan Fc defined by N. Reading. By construction, Assoc(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' two Coxeter elements in W, we have that Assoc(W) and Assoc'(W) are isometric if and only if μ(c') = c or μ(c')=w₀c⁻¹w₀ for μ an automorphism of the Coxeter graph of W such that κₛ=κμ₍ₛ₎ for all s∈ S. As a byproduct, we classify the isometric Cambrian fans of W.