发表机构
Aarhus University; University of Zagreb Faculty of Electrical Engineering and Computing（奥胡斯大学; 萨格勒布大学电气工程和计算学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推广了雷尼汽车停车问题，推导了带形区域上梯形停车的停车常数显式公式，得到有限尺寸渐近结果，发现停车常数随底边长比非单调变化且存在唯一极小值点。
AI 中文摘要
本文研究了雷尼（Rényi）汽车停车问题的一种推广，即等腰梯形被依次放置在一条带形区域上。我们推导了停车常数的显式公式，该公式用梯形的两个底边长表示，经典的矩形和三角形模型可作为特例由此得到。我们进一步得到了已放置梯形的期望数量和方差的有限尺寸渐近行为，其收敛速率明确依赖于所放置粒子的几何形状。特别地，尽管递归构造涉及两种不同的基底几何，但它们的方差具有相同的主导渐近密度。最后，我们证明停车常数对两个底边长的比值呈非单调依赖，且存在唯一的极小值点，因此效率最低的形状是真正的梯形而非三角形。
英文摘要
In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.
Comments39 pages, 8 figures
