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We define a family of convex polytopes called constrainahedra, which indexcollisions of horizontal and vertical lines. Our construction proceeds by firstdefining a poset $C(m,n)$ of good rectangular preorders, then proving that$C(m,n)$ is a lattice, and finally constructing a polytopal realization bytaking the convex hull of a certain explicitly-defined collection of points.The constrainahedra will form the combinatorial backbone of the second author'sconstruction of strong homotopy duoids. We indicate how constrainahedra couldbe realized as Gromov-compactified configuration spaces of horizontal andvertical lines; viewed from this perspective, the constrainahedra includenaturally into the first author's notion of 2-associahedra.

Constrainahedra