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Stability for binary scalar products

Andrey Kupavskii, Dmitry Tsarev

TL;DR

The paper resolves a strong version of a conjecture on 2-level polytopes by proving a tight stability bound: among $d$-dimensional $2$-level polytopes not affinely equivalent to the cube or cross-polytope, the vertex–facet product satisfies $f_0(P)f_{d-1}(P)\le (d-1)2^{d+1}+8(d-1)$. The authors extend prior binary-scalar-product bounds, develop a projection-induction framework, and carry out careful case analyses (including a detailed 3c case) to bound the product and classify extremal structures; they also corroborate the structure with small-dimension enumerations. The main contribution is a sharp stability result built on a refined understanding of the binary-inner-product configurations, yielding a near-complete dichotomy that tightens the picture beyond cube and cross-polytope cases. This advances the theory of 2-level polytopes by quantifying how far from the extremal cubes and cross-polytopes a polytope can be while nearly attaining the known upper bound, with two explicit constructions achieving the bound’s tightness.

Abstract

Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) conjectured that 2-level polytopes cannot simultaneously have many vertices and many facets, namely, that the maximum of the product of the number of vertices and facets is attained on the cube and cross-polytope. This was proved in a recent work by Kupavskii and Weltge. In this paper, we resolve a strong version of the conjecture by Bohn et al., and find the maximum possible product of the number of vertices and the number of facets in a 2-level polytope that is not affinely isomorphic to the cube or the cross-polytope. To do this, we get a sharp stability result of Kupavskii and Weltge's upper bound on $\left|\mathcal A\right|\cdot\left|\mathcal B\right|$ for $\mathcal A,\mathcal B \subseteq \mathbb R^d$ with a property that $\forall a \in \mathcal A, b \in \mathcal B$ the scalar product $\langle a, b\rangle \in\{0,1\}$.

Stability for binary scalar products

TL;DR

The paper resolves a strong version of a conjecture on 2-level polytopes by proving a tight stability bound: among -dimensional -level polytopes not affinely equivalent to the cube or cross-polytope, the vertex–facet product satisfies . The authors extend prior binary-scalar-product bounds, develop a projection-induction framework, and carry out careful case analyses (including a detailed 3c case) to bound the product and classify extremal structures; they also corroborate the structure with small-dimension enumerations. The main contribution is a sharp stability result built on a refined understanding of the binary-inner-product configurations, yielding a near-complete dichotomy that tightens the picture beyond cube and cross-polytope cases. This advances the theory of 2-level polytopes by quantifying how far from the extremal cubes and cross-polytopes a polytope can be while nearly attaining the known upper bound, with two explicit constructions achieving the bound’s tightness.

Abstract

Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) conjectured that 2-level polytopes cannot simultaneously have many vertices and many facets, namely, that the maximum of the product of the number of vertices and facets is attained on the cube and cross-polytope. This was proved in a recent work by Kupavskii and Weltge. In this paper, we resolve a strong version of the conjecture by Bohn et al., and find the maximum possible product of the number of vertices and the number of facets in a 2-level polytope that is not affinely isomorphic to the cube or the cross-polytope. To do this, we get a sharp stability result of Kupavskii and Weltge's upper bound on for with a property that the scalar product .
Paper Structure (12 sections, 12 theorems, 68 equations, 2 figures)

This paper contains 12 sections, 12 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Discussion of the proofs
  4. Preliminaries
  5. Notation.
  6. The setup from kupavskii22
  7. Auxiliary results
  8. Proofs of the main results
  9. Uniqueness for Theorem \ref{['d_plus_one_two_d']}
  10. Proof of Theorem \ref{['d2d_plus_2d']}
  11. Application to 2-level polytopes
  12. Appendix

Key Result

theorem 1

If $P$ is a $d$-dimensional 2-level polytope, its number of vertices $f_0(P)$ and facets $f_{d-1}(P)$ satisfy

Figures (2)

  • Figure 1: Possible sizes of families ${\mathscr{A}}$, ${\mathscr{B}}$ that span ${\mathbb{R}}^5$ and have binary scalar products.
  • Figure 2: $\operatorname{min}(|{\mathscr{A}}|, |{\mathscr{B}}|)$ and $|{\mathscr{A}}||{\mathscr{B}}|$ for families that span ${\mathbb{R}}^5$ and have binary scalar products.

Theorems & Definitions (44)

  • theorem 1
  • theorem 2
  • example 1: Suspension of a cube
  • example 2: Cross-polytope $\times$ segment
  • theorem 3
  • theorem 4
  • example 3
  • example 4
  • conjecture 1
  • example 5
  • ...and 34 more