Abstract A proof is given, for a large class of polygons, of a conjecture of Shier concerning his algorithm for the disjoint but otherwise random placement of successively smaller copies of an arbitrary shape within a bounded region whose area is the infinite sum of all the shape areas. If the shapes decrease successively in size according to a power law with an exponent falling in a specified range of values, dependent on the particular shape, Shier conjectured there will always be sufficient room within the bounded region to place the next shape in the sequence, disjoint from all previous shapes. The conjecture implies the shapes would fill the bounded region, except for a possible set of measure zero. Our proof confirms the conjecture for all convex polygons, as well as those nonconvex polygons that satisfy a certain condition that we call double containment. Double containment seems to us a natural condition to place on the nonconvexity of polygons and may prove of interest beyond its use in this article. Empirically, Shier’s conjecture appears true for a much larger range of exponents and variety of shapes. Some possible reasons for this are discussed.

中文翻译：

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裂缝之间：用多边形填充空间

摘要 对一大类多边形，给出了 Shier 关于他的算法的猜想的证明，该算法用于将任意形状的连续较小副本随机放置在一个有界区域内，该区域的面积是所有形状的无限和。领域。如果形状的大小根据幂律连续减小，并且指数落在指定的值范围内，取决于特定的形状，Shier 推测在有界区域内总会有足够的空间来放置序列中的下一个形状，与之前的所有形状脱节。该猜想意味着形状将填充有界区域，除了可能的测量值集为零。我们的证明证实了所有凸多边形的猜想，以及那些满足我们称为双重包含的特定条件的非凸多边形。在我们看来，双重包含似乎是多边形非凸性的自然条件，并且可能证明在本文中使用它之外的其他意义。从经验上看，Shier 的猜想似乎适用于更大范围的指数和各种形状。讨论了一些可能的原因。