A lower bound of 0.8366 for Lebesgue's universal covering problem

A universal cover is a convex set in the plane that contains a congruent copy of every set of diameter at most 1. Lebesgue asked in 1914 for a universal cover of least area; before this work, the infimum Λ of the areas of universal covers was known to satisfy 0.83474 ≤ Λ ≤ 0.8441. Lower bounds come from test sets: Λ is bounded below by the least area of a convex hull of congruent copies of given sets of diameter at most 1. The best earlier lower bounds used the disk, the Reuleaux triangle and the Reuleaux pentagon, but these three bodies are known to admit a placement with hull area below 0.83479, so they cannot give a much better bound. Adding the regular Reuleaux heptagon and nonagon of width 1, we improve the lower bound to Λ ≥ 0.8366. Without the nonagon, the other four bodies already give Λ ≥ 0.836. The proof is a computer-assisted branch and bound over the relative placements of the bodies. Its certificates are checked in interval arithmetic, and separate proofs of both bounds in Lean 4, with their own certificates, are checked entirely by Lean's kernel.

The certificate for 0.8366 is a tree of about 32 million boxes in the eleven-dimensional space of placements of the five bodies. Local minimization finds a placement of these bodies whose hull has area about 0.83730, 0.0007 above the bound proved; numerically, no lower bound from them can exceed that area. The certificates, the checker, the programs and the Lean proofs are archived at doi:10.5281/zenodo.23717912 and available at github.com/tejasvi/lebesgue-cover-bound/tree/v9.