Rigorous areas for the classical Lebesgue universal covering ladder
Abstract
Lebesgue's universal covering problem, posed in 1914, asks for the convex set of least
area containing an isometric copy of every planar set of diameter 1. The known upper
bounds descend through a ladder of constructions — the regular hexagon, Pál (1920),
Sprague (1936), Hansen (1992), Baez–Bagdasaryan–Gibbs (2015), Gibbs (2018) — whose areas
are quoted in the literature to at most twelve decimal places. So far as I can determine
none had been evaluated with rigorous error control. This note certifies the whole
ladder:
hexagon √3/2
Pál 2 − 2/√3
Sprague 0.844137708435197570894066994044…
Hansen 0.844137708397690336757182874586…
BBG 2015 0.844115376859376746806104420762… (σ = 1.3)
0.844115297128419059214192192376… (Egan's σ)
Gibbs 2018 a ≤ 0.8440935944, certified with margin 9.3×10⁻¹²
The last line is, to my knowledge, the first verification of the record since its
publication eight years ago; together with the certified lower bound of Xie (2026), the
problem's bracket 0.833 ≤ a ≤ 0.8440935944 now rests on certified computation at both
ends. Section 3 reconstructs Sprague's reduction and records a point of attribution: his
1936 paper proves only that Pál's cover is not minimal, and states no area. Section 4
describes the arithmetic. Section 5 confirms every digit of the Baez–Bagdasaryan–Gibbs
table and settles a discrepancy with a figure circulated on the Azimuth blog. Sections 6
and 7 certify the 2015 cover and the 2018 record, the latter by running the record's
five-step boundary map in second-order jet arithmetic over interval coefficients to
bound the discretisation error rigorously. Section 8 reports a soundness defect in the
kernel's first conversions — caught by a consistency gate between independently
certified rungs — with its fix and its blast radius. Universality, that each set does
cover every diameter-1 set, is taken from the literature throughout; the computational
side conditions of the 2015 construction are certified here.
Cite it
Gonzalez, V. (2026). Rigorous areas for the classical Lebesgue universal covering ladder. Zenodo. https://doi.org/10.5281/zenodo.21701519