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Generalized angle vectors, geometric lattices, and flag-angles
Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 441-464

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.484
Classification: 52B11, 52B45, 52C35, 06A11, 52B12
Keywords: interior/exterior angles, cone valuations, Gram’s relation, graded posets, angle deficiencies, flag-angles, flag-vectors, incidence algebras
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Backman, Spencer; Manecke, Sebastian; Sanyal, Raman. Generalized angle vectors, geometric lattices, and flag-angles. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 441-464. doi: 10.5802/alco.484
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