Binary scalar products

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 d2^d+1, which we show to be true.