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Register a new userGeneralized angle vectors, geometric lattices, and flag-angles
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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.
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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 extreme rays of the cone for Eulerian posets. A new family of linear inequalities valid for flag vectors of Eulerian posets is given.
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 $(alpha_1,ldots,alpha_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(x_1,dots, x_{k+2})in E^{k+2}$ satisfying $angle (x_j,x_{j+1},x_{j+2})=alpha_j$ for every $1le j le k$ as well as pinned analogues.
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, quad gamma_2leqbinom{gamma_1}{2}, quad gamma_{gamma_1}in{0,1}, quad text{ and }gamma_{gamma_1-1}in{0,1,2,gamma_1}, end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Delta$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $gamma$-vectors of flag homology spheres (of any dimension); these are the first examples of this kind. In addition, we prove a flag analog of Perles 1970 theorem on $k$-skeleta of polytopes with few vertices, specifically: the number of combinatorial types of $k$-skeleta of flag homology spheres with $gamma_1leq b$, of any given dimension, is bounded independently of the dimension.
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 prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. (2006). Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton (2003) in the theory of normal surfaces.
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 $pi$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).
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