An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound

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 ${\Lambda }_M$ satisfy $a_{\mathrm{Leb}}={\lim }_{M\to \infty }{\Lambda }_M$, and each level is a continuous finite-dimensional problem. We prove $0\le a_{\mathrm{Leb}}-{\Lambda }_M\le C{M}^{-2}$, 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 $a_{\mathrm{Leb}}\ge 0.834$, improving the lower-bound benchmark established by Brass and Sharifi in 2005.