  We show that the mesh mutations are the minimal relations among the
$\boldsymbol{g}$-vectors with respect to any initial seed in any finite type
cluster algebra. We then use this algebraic result to derive geometric
properties of the $\boldsymbol{g}$-vector fan: we show that the space of all
its polytopal realizations is a simplicial cone, and we then observe that this
property implies that all its realizations can be described as the intersection
of a high dimensional positive orthant with well-chosen affine spaces. This
sheds a new light on and extends earlier results of N. Arkani-Hamed, Y. Bai, S.
He, and G. Yan in type $A$ and of V. Bazier-Matte, G. Douville, K. Mousavand,
H. Thomas and E. Yildirim for acyclic initial seeds.
  Moreover, we use a similar approach to study the space of polytopal
realizations of the $\boldsymbol{g}$-vector fans of another generalization of
the associahedron: non-kissing complexes (a.k.a. support $\tau$-tilting
complexes) of gentle algebras. We show that the space of realizations of the
non-kissing fan is simplicial when the gentle bound quiver is brick and
$2$-acyclic, and we describe in this case its facet-defining inequalities in
terms of mesh mutations.
  Along the way, we prove algebraic results on $2$-Calabi-Yau triangulated
categories, and on extriangulated categories that are of independent interest.
In particular, we prove, in those two setups, an analogue of a result of M.
Auslander on minimal relations for Grothendieck groups of module categories.
