发表机构
Chongqing University of Posts and Telecommunications（重庆邮电大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对1914年提出的勒贝格通用覆盖问题，构建精确勒洛型变分层次结构，证明其下界不小于0.834，改进了2005年的基准下界。
AI 中文摘要
勒贝格通用覆盖问题由勒贝格于1914年提出，旨在寻找能包含所有直径不超过1的集合的全等副本的最小面积平面凸集。我们为该常数引入了精确的勒洛型变分层次结构：其单调有限弧值Λ_M满足a_Leb=lim_{M→∞}Λ_M，且每个层级均为连续有限维问题。我们证明0≤a_Leb−Λ_M≤C M⁻²，为逼近该常数提供了可控的有限弧路径。作为经认证的低阶实现，针对正则有限勒洛子测试的外圆角区间证书证明a_Leb≥0.834，改进了Brass与Sharifi于2005年建立的下界基准。
英文摘要
Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values $Λ_M$ satisfy $a_{\mathrm{Leb}}=\lim_{M\to\infty}Λ_M$, and each level is a continuous finite-dimensional problem. We prove $0\le a_{\mathrm{Leb}}-Λ_M\le C M^{-2}$, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves $a_{\mathrm{Leb}}\ge0.834$, improving the lower-bound benchmark established by Brass and Sharifi in 2005.
