This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

本文是通过处理现实生活应用程序的需求，使公平分配理论更接近实践的持续努力的一部分。我们关注源自土地划分的两个要求：(1) 每个代理人都应收到一块可用几何形状的地块，以及 (2) 不同代理人的地块必须在物理上分开。有了这些要求，比例性的经典公平概念就不切实际了，因为可能无法获得它的任何乘法近似值。相反，序数最大最小份额近似，由 Budish 于 2011 年推出，提供有意义的公平性保证。我们证明了当可用形状是正方形、胖矩形或任意轴对齐矩形时可实现的最大最小份额保证的上限和下限，并探讨了在此设置中寻找公平分区的算法和查询复杂性。我们的工作利用了来自计算几何的工具和概念，例如独立的矩形集和断头台分区。