Andrey Kupavskii:

Binary scalar products
 
Abstract: 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.