Heilbronn's triangle problem asks: place n points in a unit square to maximize the minimum area of any triangle formed by three of those points. Spread the points as evenly as possible, but every triple must be considered. The optimal configuration for n = 9 was found in 2002 by Comellas and Yebra. It was not proven optimal until now.
Sudermann-Merx (arXiv:2603.11107) certifies global optimality of the 2002 configuration using mixed-integer nonlinear programming combined with exact symbolic computation. The computation takes fifteen minutes on a standard desktop. The paper also derives exact coordinates — not floating-point approximations but algebraic numbers — for all optimal configurations from n = 5 through n = 9.
The structural interest is in where the difficulty lived. Finding the optimal configuration for n = 9 was done twenty-two years ago by heuristic search. The configuration was known. What was not known was whether anything better existed. Proving that it did not required exhausting all alternatives — not just searching the space efficiently but certifying that no unexplored region could contain a superior arrangement. Heuristic search finds good points. Global optimization proves there are no better ones. These are different computational acts.
The gap between discovery and certification is characteristic of extremal geometry. For small n, the optimal configurations are often guessable — symmetry and intuition narrow the search space — but the space of all possible placements is continuous and high-dimensional. A configuration that looks optimal under local perturbation might be a local maximum. Proving global optimality requires either an exhaustive argument (which scales poorly) or a dual bound (which requires reformulating the problem in a way that upper-bounds the objective). Sudermann-Merx introduces a symmetry-breaking strategy that reduces the search space and exploits structural properties of determinants — the area of a triangle is half the absolute value of a 3×3 determinant — to tighten the relaxation.
The result also reveals structure in the noncritical triangles. In the optimal configuration, the areas of triangles that are not the minimum cluster around distinct values rather than spreading uniformly. The extremal configuration organizes the non-extremal geometry. The constraint on the worst case shapes the typical case — not just preventing small triangles but inducing regularity in the rest.
Twenty-two years between finding and proving. The answer was not the bottleneck. The proof was.