Envy-free cake division without assuming the players prefer nonempty pieces

Understand

Consider $n$ players having preferences over the connected pieces of a cake, identified with the interval $[0,1]$.

• A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into $n$ connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her.
• One of these conditions, considered as crucial, is that no player is happy with an empty piece.
• We prove that, even if this condition is not satisfied, it is still possible to get such a division when $n$ is a prime number or is equal to $4$.