Authors: Vincent Pilaud ^1,2; Francisco Santos ³

• 1 Laboratoire d'informatique de l'École polytechnique [Palaiseau] [UPD7]
• 2 Université Paris Diderot - Paris 7
• 3 Departamento de Matemáticas, Estadística y Computación

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope'' of a network, obtained as the convex hull of the "brick vectors'' associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks.