Constrainahedra

Abstract

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