  In this article we study the problem of fair division. In particular we study
a notion introduced by J. Barbanel that generalizes super envy-free fair
division. We give a new proof of his result. Our approach 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 inequalities constraints? Furthermore, when all the measures
are given with piecewise constant density functions then we show how to
construct effectively such a fair division.
