Generalized angle vectors, geometric lattices, and flag-angles

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--Poincar\'e relation as the unique linear relation among interior angles. We show the existence and uniqueness of Euler--Poincar\'e-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.