Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for $n \le 8$
Year: 2026
Type: preprint
Abstract: We study Heilbronn's triangle problem in the unit right triangle, where $n$ points are placed to maximize the smallest of the $\binom{n}{3}$ triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with $n \ge 5$ has at least four points on the boundary, one edge containing two of them. With the affine $S_3$ symmetry this fixes four boundary points and $n$ orientation variables in a mixed-integer model that certifies global optimality for all $n \le 8$ up to a relative gap of $10^{-4}$. For $n \le 7$ this independently confirms the known exact optima, the case $n = 7$ having been proved by Zeng and Chen with symbolic computation. For $n = 8$, where only bounds were known, it narrows the gap to Cantrell's configuration fourfold in 39 minutes on one machine. The optimum there 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 $S_7$, so on that conjecture no expression in radicals exists.
Source: arXiv (Cornell University)
Author: Sudermann-Merx, Nathan
Language: English
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169509344