等宽曲线与勒贝格覆盖问题
Curves of constant width and Lebesgue's covering problem
- Indian Institute of Information Technology Una, India（印度乌纳信息技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对勒贝格覆盖问题，以等宽曲线为测试集，证明凸通用覆盖集面积下界为0.8344，改进了此前结果，相关证明含解析归约与有限计算。
AI中文摘要：
通用覆盖集是平面上的凸集，可包含所有直径为1的平面集合的全等副本。勒贝格于1914年提出：求面积最小的通用覆盖集，但其具体值至今未知。本文证明，每个凸通用覆盖集的面积至少为0.8344，改进了2005年发表的0.832，以及2026年预印本给出的0.833，后两者均来自圆盘、等边三角形与正五边形的组合。本文的测试集为等宽曲线：圆盘、勒洛三角形与勒洛五边形，每种等宽曲线都包含其构建所基于的正多边形，因此在相同数量的集合和相同放置参数下，该系列的覆盖范围更广泛；研究表明，经典配置存在一种排列，其凸包面积低于0.8336，因此通过该论证从这三个集合得出的任何界都无法达到本文的结果。等宽曲线曾由吉布斯于2014年提出用于该用途并进行了数值探索，本文新增的内容是相关证明，该证明包含解析归约和一次有限计算：归约通过将每个勒洛多边形侵蚀为固定集合（该集合包含在该放置框允许的所有放置中），一次性在整个放置框上对凸包面积进行下界估计；计算为对所得五维空间的穷小子分，记录为含486799600个节点的证书，由独立于搜索的验证器检查，其浮点误差的严格界比验证达到的余量小数千倍。
英文摘要：
A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.
