Valid Orderings of Real Hyperplane Arrangements

TL;DR

This paper introduces the valid order arrangement for real hyperplane arrangements, linking geometric crossing orders to combinatorial structures, and explores applications to polytopes, shellings, and chromatic polynomials.

Contribution

It defines the valid order arrangement and connects it to the Dilworth truncation, providing new insights into hyperplane crossing orders and their applications.

Findings

Maximum line shellings of a polytope with fixed point p determined

Valid order arrangements relate to Dilworth truncation of the semicone of A

Generalization of chromatic polynomials for order polytopes

Abstract

Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(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 A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of p where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the *Dilworth truncation* of the semicone of 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 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.