发表机构
Hangzhou Normal University; Cardiff University; Nanjing Normal University（杭州师范大学; 卡迪夫大学; 南京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了覆盖因子超过三成员尖锐界的有限不可分离正位似三角形族，否定Bezdek与Lángi的猜想，并给出显式例子、循环递推及连续模型候选值。
AI 中文摘要
平面凸体的一个有限族被称为不可分离族，如果不存在一条与它们的并集不相交的直线，使得该直线的每个开半平面中至少包含该族中的一个成员。在本文中，我们证明了存在具有正位似三角形的有限不可分离族，其覆盖因子严格超过尖锐的三成员界 $\mu = \frac{2}{3} + \frac{2}{3\sqrt{3}}$。这否定了由 K. Bezdek 和 Z. Lángi 提出的一个问题，他们最初在证明 A. W. Goodman 和 R. E. Goodman 关于圆盘的经典因子 1 覆盖定理对任意正位似体不成立之后，建立了这个三成员界。除了给出具有四个、五个和六个成员且因子分别约为 1.0533161、1.0551900 和 1.0572061 的显式代数例子外，我们还提供了一个共同的循环递推，该递推产生一个具有精确因子 250000000/235141779 的 303 成员族。最后，我们推导出一个由越来越精细的递推所提示的连续模型，得到一个数值候选值 1.0633083；其可达性和最优性仍然开放。
英文摘要
A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound $μ= \frac{2}{3} + \frac{2}{3\sqrt{3}}$. This resolves negatively a question posed by K. Bezdek and Z. Lángi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.
Comments22 pages, 4 figures
