Valid Orderings of Real Hyperplane Arrangements
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454_2015_Article_9683.pdf
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481.66 KB
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626438bcf340ce5aa02ae2d98fe7c9d4
Author(s)
Stanley, Richard P
Date Issued
April 2015
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Stanley, Richard P. “Valid Orderings of Real Hyperplane Arrangements.” Discrete & Computational Geometry 53.4 (2015): 951–964.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00454-015-9683-0
