Valid Orderings of Real Hyperplane Arrangements

Abstract

Given a real finite hyperplane arrangement $\mathcal {A}$ and a point $p$ not on any of the hyperplanes, we define an arrangement $\mathrm {vo}(\mathcal {A},p)$, called the valid order arrangement, whose regions correspond to the different orders in which a line through $p$ can cross the hyperplanes in $\mathcal {A}$. If $\mathcal {A}$ is the set of affine spans of the facets of a convex polytope $\mathcal {P}$ and $p$ lies in the interior of $\mathcal {P}$, then the valid orderings with respect to $p$ are just the line shellings of $\mathcal {P}$ where the shelling line contains $p$. When $p$ is sufficiently generic, the intersection lattice of $\mathrm {vo}(\mathcal {A},p)$ is the Dilworth truncation of the semicone of $\mathcal {A}$. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a $d$-polytope with $m$ facets when the shelling line contains a fixed point $p$. If $\mathcal {P}$ is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.