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Generalized angle vectors, geometric lattices, and flag-angles

Spencer Backman, Sebastian Manecke, Raman Sanyal

Abstract

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.

Generalized angle vectors, geometric lattices, and flag-angles

Abstract

Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of -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--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--vectors of graded posets. This allows us to determine the linear relations satisfied by interior/exterior flag-angle vectors.

Paper Structure

This paper contains 7 sections, 34 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Cone angles and linear relations
  3. Belt polytopes and angle vectors
  4. Connecting angles with Möbius inversion
  5. Flag-angle vectors
  6. Flag-Whitney numbers and zonotopes
  7. Spherical intrinsic volumes and Grassmann angles

Key Result

Theorem 1.1

Let $\alpha$ be a cone angle. Then, up to scaling, the only linear relations satisfied by $\widehat{\alpha}(P)$, respectively $\widecheck{\alpha}(P)$, for any $d$-dimensional polytope $P$ are

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Volland
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4: AS15
  • ...and 49 more