2026/07/16 by Nathan Sudermann-Merx
#math.OC #math.CO
We study Heilbronn's triangle problem in the unit right triangle, where n points are placed to maximize the smallest of the \binomn3 triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with n ≥ 5 has at least four points on the boundary, one edge carrying two of them. With the affine S3 symmetry this fixes four boundary points and n orientation variables in a mixed-integer model that certifies global optimality for all n ≤ 8: for n = 8 apparently the first proof, and for n = 7 an independent confirmation of the symbolic-computation proof of Zeng and Chen. For n ≤ 7 we obtain exact optima with explicit configurations. For n = 8 the optimum is conjectured to be the real root of a septic obtained by Chen, Zeng and Zhou, which our reconstruction confirms to 250 digits. We show its Galois group is S7, so on that conjecture no expression in radicals exists.