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Proceedings of the American Mathematical Society

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Coarsening polyhedral complexes

Author: Nathan Reading
Journal: Proc. Amer. Math. Soc. 140 (2012), 3593-3605
MSC (2010): Primary 52B99, 52C35
Published electronically: February 23, 2012
MathSciNet review: 2929028
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Abstract: Given a pure, full-dimensional, locally strongly connected polyhedral complex  $ \mathcal {C}$, we characterize, by a local codimension-$ 2$ condition, polyhedral complexes that coarsen  $ \mathcal {C}$. The proof of the characterization draws upon a general shortcut for showing that a collection of polyhedra is a polyhedral complex and upon a property of hyperplane arrangements which is equivalent, for Coxeter arrangements, to Tits' solution to the Word Problem. The motivating special case, the case where  $ \mathcal {C}$ is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietze's convexity theorem.

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Nathan Reading
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

Received by editor(s): May 6, 2010
Received by editor(s) in revised form: April 14, 2011
Published electronically: February 23, 2012
Additional Notes: The author was partially supported by NSA grant H98230-09-1-0056.
Communicated by: Jim Haglund
Article copyright: © Copyright 2012 Nathan Reading