How to cut a cake with a Gram matrix

In this article we study a problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We call this notion hyper envy-free. We give a new proof for the existence of such fair divisions. Our approach relies on classical linear algebra tools and allows us to give an explicit bound for this kind of fair division. Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequality constraints? Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division.