  We generalize the brick polytope of V. Pilaud and F. Santos to spherical
subword complexes for finite Coxeter groups. This construction provides
polytopal realizations for a certain class of subword complexes containing all
cluster complexes of finite types. For the latter, the brick polytopes turn out
to coincide with the known realizations of generalized associahedra, thus
opening new perspectives on these constructions. This new approach yields in
particular the vertex description of generalized associahedra, a Minkowski sum
decomposition into Coxeter matroid polytopes, and a combinatorial description
of the exchange matrix of any cluster in a finite type cluster algebra.
