Binary scalar products

TL;DR

This paper proves a bound on the size of sets with binary scalar products and confirms a conjecture relating vertices and facets of 2-level polytopes, advancing understanding in polytope theory.

Contribution

It establishes a new upper bound on the product of set sizes with binary scalar products and confirms a conjecture on 2-level polytopes' vertices and facets.

Findings

Bound |A|·|B| ≤ (d+1) 2^d for sets with binary scalar products.

Confirmed the conjecture that vertices·facets ≤ d·2^{d+1} for 2-level polytopes.

Provided a key inequality that settles a conjecture in polytope theory.

Abstract

Let $A,B \subseteq \mathbb{R}^d $ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $ |A| \cdot |B| \le (d+1) 2^d $. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane $H$ there is a parallel hyperplane $H'$ such that $H \cup H'$ contain all vertices. The authors conjectured that for every $d$-dimensional 2-level polytope $P$ the product of the number of vertices of $P$ and the number of facets of $P$ is at most $d 2^{d+1}$, which we show to be true.