Lower bounds for the Beardwood–Halton–Hammersley constant

Let β be the Beardwood–Halton–Hammersley constant: the shortest tour through n independent uniform random points in the unit square has length asymptotic to β√n almost surely. Dong and Cao proved β ≥ 0.6421 from an inequality β ≥ 5/8 + ∑_k=3^K C_k, where C_k is an integral over configurations of k points in the plane that form a connected graph when points at distance at most 1 are joined. For C₃, C₄ and C₅, sharper bounds come from a first-order expansion of the area of the union of the unit disks on boxes of configurations. For C₆ + … + C₂₂, which Dong and Cao did not use, a lower bound comes from comparing each point only with the three points before it in the order of their first coordinates. We obtain β ≥ 0.6496, and β ≥ 0.6508 if the exponential function of the C library used to evaluate the bound for C₆ + … + C₂₂ has a relative error below 10⁻⁹. The proofs are computer-assisted; the bound 0.6496 is proved in Lean 4, with finite checks evaluated by compiled code, from the inequality of Dong and Cao and two statements about unions of disks that we prove but do not formalize.

The programs, logs and the Lean formalization are archived at doi:10.5281/zenodo.23716095 and available at github.com/tejasvi/bhh-constant-bound-v8/tree/v9 (commit eeb68cd603d2).