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Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes

Raman Sanyal, Martin Winter

TL;DR

The paper proves Kalai's $3^d$ conjecture for two notable classes of polytopes: unconditional and locally anti-blocking. It introduces the notions of special points and a coordinate-geometry graph $G_P$ to classify minimizers, showing that equality in the $s(P)\ge 3^d$ bound occurs precisely for (generalized) Hanner polytopes via a halfspace-scaling equivalence. A key methodological contribution is reconstructing a minimizer from $G_P$, and establishing a complete combinatorial characterization of minimizers through cographs, supported by a second, purely combinatorial proof for unconditional polytopes. The work connects Kalai's conjecture to the Mahler framework and opens avenues on combinatorial realizations and the flag conjecture within the locally anti-blocking setting.

Abstract

Kalai's $3^d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3^d$ faces. We give short proofs for two special cases: if $P$ is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if $P$ is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.

Kalai's $3^{d}$ conjecture for unconditional and locally anti-blocking polytopes

TL;DR

The paper proves Kalai's conjecture for two notable classes of polytopes: unconditional and locally anti-blocking. It introduces the notions of special points and a coordinate-geometry graph to classify minimizers, showing that equality in the bound occurs precisely for (generalized) Hanner polytopes via a halfspace-scaling equivalence. A key methodological contribution is reconstructing a minimizer from , and establishing a complete combinatorial characterization of minimizers through cographs, supported by a second, purely combinatorial proof for unconditional polytopes. The work connects Kalai's conjecture to the Mahler framework and opens avenues on combinatorial realizations and the flag conjecture within the locally anti-blocking setting.

Abstract

Kalai's conjecture states that every centrally-symmetric -polytope has at least faces. We give short proofs for two special cases: if is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.
Paper Structure (12 sections, 10 theorems, 15 equations, 4 figures)

This paper contains 12 sections, 10 theorems, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Basic facts about locally anti-blocking polytopes
  4. The inequality part of \ref{['thm:main_lab']}
  5. The minimizer part of \ref{['thm:main_lab']}
  6. Halfspace scaling
  7. Reconstructing $\boldsymbol{P}$ from $\boldsymbol{G_P}$.
  8. Characterizing Hanner polytopes
  9. A combinatorial proof of \ref{['cor:main_unconditional']}
  10. Open questions
  11. Combinatorially locally anti-blocking
  12. The flag conjecture

Key Result

Theorem 3

If $P \subset \mathbb{R}^d$ is an unconditional polytope, then $s(P) \ge 3^d$. Moreover, $s(P) = 3^d$ if and only if $P$ is a Hanner polytope.

Figures (4)

  • Figure 1: Visualization of two unconditional polytopes, a locally anti-blocking polytope that is not unconditional, and a polytope that is neither even though it is centrally-symmetric.
  • Figure 2: An unconditional polytope $P$ and a section $P\cap H$ (red) with a coordinate hyperplane that "exposes" more than a third of the faces.
  • Figure 3: An axis-aligned quadrangle (left) and a diamond (right).
  • Figure 4: The 3-dimensional minimizers $P$ and their associa-ted graphs $G_P$. The shaded areas visualize intersections of $P$ with the 2-dimensio-nal coordinate subspaces. It holds the more general observation that for an axis-aligned cube the graph $G_P$ is complete, and for an "axis-aligned" crosspolytope the graph $G_P$ is edge-less.

Theorems & Definitions (32)

  • Conjecture 1: $3^d$ conjecture
  • Conjecture 2: Mahler conjecture
  • Theorem 3
  • Theorem 4
  • Remark 5: Zonotopes
  • Remark 6: Inductive approach
  • Lemma 7: see ArtsteinSadovskySanyal
  • proof
  • Lemma 9
  • proof
  • ...and 22 more