An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Zeng, Shuai
Institution: Chongqing University of Posts and Telecommunications
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206160608