English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Consider $n$ players having preferences over the connected pieces of a cake,identified with the interval $[0,1]$. A classical theorem, found independentlyby Stromquist and by Woodall in 1980, ensures that, under mild conditions, itis possible to divide the cake into $n$ connected pieces and assign thesepieces to the players in an envy-free manner, i.e, such that no player strictlyprefers 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 provethat, even if this condition is not satisfied, it is still possible to get sucha division when $n$ is a prime number or is equal to $4$. When $n$ is at most$3$, this has been previously proved by Erel Segal-Halevi, who conjectured thatthe result holds for any $n$. The main step in our proof is a new combinatoriallemma in topology, close to a conjecture by Segal-Halevi and which isreminiscent of the celebrated Sperner lemma: instead of restricting the labelsthat can appear on each face of the simplex, the lemma considers labelings thatenjoy a certain symmetry on the boundary.

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