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Kalai's flag conjecture for locally anti-blocking polytopes

Arnon Chor

TL;DR

The paper proves Kalai's full flag conjecture for the class of locally anti-blocking polytopes by establishing a tight lower bound of $2^d\cdot d!$ flags for every $d$-dimensional proper locally anti-blocking polytope and characterizing equality precisely by generalized Hanner polytopes. The approach provides an elementary inductive proof based on a sign decomposition of flags via the standard fan, a flip (monodromy) mechanism, and an injective lifting strategy that raises flags along facets. A key ingredient is the normalization step that reduces to normalized polytopes and the graph-theoretic framework involving $\mathcal{G}_P$, duality, and joins, which together yield a complete equality case analysis. Overall, the work extends Kalai's flag conjecture to locally anti-blocking polytopes, complements previous results on special subclasses, and offers a transparent, combinatorial route to identifying Hanner polytopes as the extremal minimizers.

Abstract

We prove Kalai's full flag conjecture for the class of locally anti-blocking polytopes, and show that there is equality if and only if the polytope is a (generalized) Hanner polytope.

Kalai's flag conjecture for locally anti-blocking polytopes

TL;DR

The paper proves Kalai's full flag conjecture for the class of locally anti-blocking polytopes by establishing a tight lower bound of flags for every -dimensional proper locally anti-blocking polytope and characterizing equality precisely by generalized Hanner polytopes. The approach provides an elementary inductive proof based on a sign decomposition of flags via the standard fan, a flip (monodromy) mechanism, and an injective lifting strategy that raises flags along facets. A key ingredient is the normalization step that reduces to normalized polytopes and the graph-theoretic framework involving , duality, and joins, which together yield a complete equality case analysis. Overall, the work extends Kalai's flag conjecture to locally anti-blocking polytopes, complements previous results on special subclasses, and offers a transparent, combinatorial route to identifying Hanner polytopes as the extremal minimizers.

Abstract

We prove Kalai's full flag conjecture for the class of locally anti-blocking polytopes, and show that there is equality if and only if the polytope is a (generalized) Hanner polytope.

Paper Structure

This paper contains 10 sections, 13 theorems, 20 equations, 5 figures.

Table of Contents

  1. Introduction and main theorem
  2. Acknowledgements
  3. Notations and preliminaries
  4. Sketch of the proof and plan of the paper
  5. Flags of polytopes and their signs
  6. A flip lemma
  7. Some properties of locally anti-blocking polytopes
  8. The inequality part of the main theorem
  9. Equality case
  10. Flag number calculation for $\mathcal{C}(\Pi_3)$

Key Result

Theorem 1.1

Any $d$-dimensional proper locally anti-blocking polytope has at least $2^d \cdot d!$ flags, with equality if and only if the polytope is a generalized Hanner polytope.

Figures (5)

  • Figure 1: Two examples of some flag $F$ (in green) and $\chi_C^D F$ (in blue), where $C$ is the cone spanned by $x,y$ and $D$ is spanned by $x,y,z$.
  • Figure 2: The cones $C,D$ intersecting at a plane; a point $x \in \mathop{\mathrm{relint}}\nolimits F_2 \cap C$ (in red); a point $y \in \mathop{\mathrm{relint}}\nolimits F_2 \cap D$ (in blue); and points $(x_j)_{j=0}^{1}$ (in purple).
  • Figure 3: The flags $F, F^i, ..., F^j$, represented by lines passing through their respective faces.
  • Figure 4: A flag $F \in {🏳^{C}}(P)$ (in green), and the $0,1$ faces of $\chi_C^D(F) \in {🏳^{D}}(P)$, in blue, where $C$ is spanned by $x$ and $D$ by $x,y$. The face $H_1$ equals $P$ itself here.
  • Figure 5: The dual facet for vertices in the first and third columns.

Theorems & Definitions (40)

  • Conjecture 1: $3^d$ conjecture
  • Conjecture 2: Mahler's conjecture
  • Conjecture 3: Kalai's (full) flag conjecture
  • Theorem 1.1
  • Theorem 1.2
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Definition 1
  • ...and 30 more