2018/09/04 by Backman, Spencer, Manecke, Sebastian, Sanyal, Raman
#06A11 #52B12 #52B45 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1809.00956
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.