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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, Grams relation takes the place of the Euler-Poincare relation as the unique linear relation among interior angles. We show the existence and uniqueness of Euler-Poincare-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.
The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extrem
We study a variant of the ErdH os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles $(alpha_1,ldots,a
We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Delta$, we show that its $gamma$-vector $gamma^Delta=(1,gamma_1,gamma_2,ldots)$ satisfies: begin{align*} gamma_j=0,text{ for all } j>gamma_1,
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to pr
We classify all sets of nonzero vectors in $mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $pi$, sol