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Lower Bounds on Face Numbers of Polytopes with $m$ Facets

Joshua Hinman

Abstract

Let $P$ be a convex $d$-polytope and $0 \leq k \leq d-1$. In 2023, this author proved the following inequalities, resolving a question of Bárány: \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr]. \] We show that for any fixed $d$ and $k$, these are the tightest possible linear bounds on $f_k(P)$ in terms of $f_0(P)$ or $f_{d-1}(P)$. We then give a stronger bound on $f_k(P)$ in terms of the Grassmann angle sum $γ_k^2(P)$. Finally, we prove an identity relating the face numbers of a polytope with the behavior of its facets under a fixed orthogonal projection of codimension two.

Lower Bounds on Face Numbers of Polytopes with $m$ Facets

Abstract

Let be a convex -polytope and . In 2023, this author proved the following inequalities, resolving a question of Bárány: \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr]. \] We show that for any fixed and , these are the tightest possible linear bounds on in terms of or . We then give a stronger bound on in terms of the Grassmann angle sum . Finally, we prove an identity relating the face numbers of a polytope with the behavior of its facets under a fixed orthogonal projection of codimension two.
Paper Structure (10 sections, 11 theorems, 38 equations)

This paper contains 10 sections, 11 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polytopes and Their Faces
  4. Solid Angles
  5. Grassmann Angles
  6. Proof of Tightness
  7. The Search for Stronger Bounds
  8. Grassmann Angles and Nonlinear Bounds
  9. Face Number Relations from a Fixed Projection
  10. Acknowledgements

Key Result

Theorem 1.1

Let $P$ be a $d$-polytope, and suppose $0 \leq k \leq d-1$. Then In the former, equality holds precisely when $k=0$ or when $k=1$ and $P$ is simple. In the latter, equality holds precisely when $k=d-1$ or when $k=d-2$ and $P$ is simplicial.

Theorems & Definitions (25)

  • Theorem 1.1: Hinman
  • Conjecture 1.2: Generalized Upper Bound Conjecture
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 15 more