In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition. We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free partition problems when $n$ is odd and not a prime power.

中文翻译：

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使用映射度的无嫉妒划分

在本文中，我们研究无羡慕分裂问题。David Gale 使用的一些此类问题的经典方法简化为考虑单纯形到自身的连续映射，并在该映射到达单纯形中心时找到充分条件。对于这样的结论，仅仅连续性是不够的，通常的假设（例如，在 Knaster--Kuratowski--Mazurkiewicz 和 Gale 定理中）是某个边界条件。我们遵循 Erel Segal-Halevi、Fr\'ed\'eric Meunier 和 Shira Zerbib，将边界条件替换为另一个假设，该假设具有经济意义的可能性，即玩家可能更喜欢细分问题中的空部分。当划分该段的玩家数量 $n$ 是素数时，我们会积极地解决问题，我们为每个不是素数的 $n$ 提供反例。当 $n$ 是奇数而不是质数时，我们还提供了与更广泛的公平或无嫉妒分区问题相关的反例。