Coarsening polyhedral complexes

Reading, Nathan

doi:10.1090/S0002-9939-2012-11194-3

Coarsening polyhedral complexes
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Proc. Amer. Math. Soc. 140 (2012), 3593-3605

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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