Generalized angle vectors, geometric lattices, and flag-angles
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866916412133474304 |
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| author | Backman, Spencer Manecke, Sebastian Sanyal, Raman |
| author_facet | Backman, Spencer Manecke, Sebastian Sanyal, Raman |
| contents | Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincaré relation as the unique linear relation among interior angles. We show the existence and uniqueness of Euler--Poincaré-type relations for generalized angle vectors by building a bridge to the algebraic combinatorics of geometric lattices, generalizing work of Klivans--Swartz.
We introduce flag-angles of polytopes as a geometric counterpart to flag-$f$-vectors. Flag-angles generalize the angle deficiencies of Descartes--Shephard, Grassmann angles, and spherical intrinsic volumes. Using the machinery of incidence algebras, we relate flag-angles of zonotopes to flag-$f$-vectors of graded posets. This allows us to determine the linear relations satisfied by interior/exterior flag-angle vectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_00956 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Generalized angle vectors, geometric lattices, and flag-angles Backman, Spencer Manecke, Sebastian Sanyal, Raman Combinatorics Metric Geometry 52B45, 52C35, 06A11, 52B12 Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincaré relation as the unique linear relation among interior angles. We show the existence and uniqueness of Euler--Poincaré-type relations for generalized angle vectors by building a bridge to the algebraic combinatorics of geometric lattices, generalizing work of Klivans--Swartz. We introduce flag-angles of polytopes as a geometric counterpart to flag-$f$-vectors. Flag-angles generalize the angle deficiencies of Descartes--Shephard, Grassmann angles, and spherical intrinsic volumes. Using the machinery of incidence algebras, we relate flag-angles of zonotopes to flag-$f$-vectors of graded posets. This allows us to determine the linear relations satisfied by interior/exterior flag-angle vectors. |
| title | Generalized angle vectors, geometric lattices, and flag-angles |
| topic | Combinatorics Metric Geometry 52B45, 52C35, 06A11, 52B12 |
| url | https://arxiv.org/abs/1809.00956 |