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首页> 外文期刊>Israel Journal of Mathematics >Envy-free cake division without assuming the players prefer nonempty pieces
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Envy-free cake division without assuming the players prefer nonempty pieces

机译：不假设球员更喜欢非空的碎片而无嫉妒的蛋糕师

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摘要

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. When n is at most 3, this has been previously proved by Erel Segal- Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.
机译：考虑n偏好于蛋糕的连接部分的偏好，用间隔[0,1]识别。一个经典的定理，由Stomquist和1980年独立发现，确保在温和的条件下，可以将蛋糕分成N连接的作品并以无嫉妒的方式分配这些件到玩家，即没有玩家严格更喜欢没有分配给她的作品。这些条件之一，被认为是至关重要的，是没有玩家对空作品感到满意。我们证明，即使不满足这种情况，当n是素数或等于4时，仍然可以获得这样的划分。当n最多3时，它之​​前被erel segal-halevi证明了这一点据劝告结果持有任何n。我们证据中的主要步骤是拓扑中的一个新的组合引理，靠近Segal-Halevi的猜想，并让人想起庆祝的尖锐物引理：而不是限制出现在Simplex的每张面上的标签，而LEMMA考虑在边界上享受某种对称性的贴标。

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