Certified upper bounds for Fejes Tóth's point-goalie problem at n = 4 and 5

Abstract

In 1974 László Fejes Tóth posed the following problem: place n points in the plane so asto minimise the largest distance from a line meeting the unit-radius disc to the nearestpoint. Writing r_n for the optimum, he proved r_1 = r_2 = 1 and r_3 = 3/5 exactly, gaveconstructions showing r_4 ≤ 0.471…, r_5 ≤ 0.406… and r_6 ≤ 1/3, and wrote that r_n isunknown for n > 3. The problem was later named the point goalie problem; its modernliterature treats the asymptotic and density regimes, and I am not aware of any publishedimprovement of the finite-n values. This note certifies, in exact rational intervalarithmetic, r_4 ≤ 0.468672 and r_5 ≤ 0.394954, improving the 1974 bounds by 0.0023 and0.0106 respectively. The certifying configurations are given by exact rationalcoordinates, and the certifier is validated by negative controls against Fejes Tóth'sproven value r_3 = 3/5. For n = 6 an unstructured search converged back to Fejes Tóth'sown configuration and produced no improvement; that is reported as a negative result, notas evidence of optimality. Finally, Fejes Tóth's skeleton argument applied to thecertified opaque barrier of length 4.799849374678… (10.5281/zenodo.21701081) giveslim sup n·r_n ≤ 2.39992468…, improving the asymptotic upper bound (π + √3)/2 = 2.43682…stated in his paper; the best known asymptotic lower bound, due to Richardson and Shepp,is 1.001. No optimality is claimed for anything presented here.