
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 both span such that holds for all , . We show that . 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 there is a parallel hyperplane such that contain all vertices. The authors conjectured that for every -dimensional 2-level polytope the product of the number of vertices of and the number of facets of is at most , which we show to be true.
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