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 as
to minimise the largest distance from a line meeting the unit-radius disc to the nearest
point. Writing r_n for the optimum, he proved r_1 = r_2 = 1 and r_3 = 3/5 exactly, gave
constructions showing r_4 ≤ 0.471…, r_5 ≤ 0.406… and r_6 ≤ 1/3, and wrote that r_n is
unknown for n > 3. The problem was later named the point goalie problem; its modern
literature treats the asymptotic and density regimes, and I am not aware of any published
improvement of the finite-n values. This note certifies, in exact rational interval
arithmetic, r_4 ≤ 0.468672 and r_5 ≤ 0.394954, improving the 1974 bounds by 0.0023 and
0.0106 respectively. The certifying configurations are given by exact rational
coordinates, and the certifier is validated by negative controls against Fejes Tóth's
proven value r_3 = 3/5. For n = 6 an unstructured search converged back to Fejes Tóth's
own configuration and produced no improvement; that is reported as a negative result, not
as evidence of optimality. Finally, Fejes Tóth's skeleton argument applied to the
certified opaque barrier of length 4.799849374678… (10.5281/zenodo.21701081) gives
lim 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.
Cite it
Gonzalez, V. (2026). Certified upper bounds for Fejes Tóth's point-goalie problem at n = 4 and 5. Zenodo. https://doi.org/10.5281/zenodo.21729548