Coarsening polyhedral complexes
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arXiv
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| Format: | Preprint |
| Publié: |
2010
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| _version_ | 1866916012152061952 |
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| author | Reading, Nathan |
| author_facet | Reading, Nathan |
| contents | Given a pure, full-dimensional, locally strongly connected polyhedral complex C with convex support, we characterize, by a local codimension-2 condition, polyhedral complexes that coarsen C. The proof of the characterization draws upon a surprising 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 C is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. The proof of the main result also implies a special case of Tietze's convexity theorem. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietze's convexity theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1004_4194 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Coarsening polyhedral complexes Reading, Nathan Combinatorics 52B99, 52C35 Given a pure, full-dimensional, locally strongly connected polyhedral complex C with convex support, we characterize, by a local codimension-2 condition, polyhedral complexes that coarsen C. The proof of the characterization draws upon a surprising 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 C is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. The proof of the main result also implies a special case of Tietze's convexity theorem. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietze's convexity theorem. |
| title | Coarsening polyhedral complexes |
| topic | Combinatorics 52B99, 52C35 |
| url | https://arxiv.org/abs/1004.4194 |