Binary scalar products

08/17/2020

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by   Andrey Kupavskii, et al.

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Let A,B ⊆ℝ^d both span ℝ^d such that ⟨ a, b ⟩∈{0,1} holds for all a ∈ A, b ∈ B. We show that |A| · |B| ≤ (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 ∪ 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.