Andrey Kupavskii: Binary scalar products

Let A,B be two families of vectors in R^d that both span it and such that $<a,b>$ is either 0 or 1 for any a, b from A and B, respectively. We show that $|A||B|\le (d+1)2^d$. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes (polytopes such that for every facet-defining hyperplane H there is its translate H' such that H together with H' cover 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. Joint work with Stefan Weltge.