Heilbronn problem in a fixed bounding box

The problem

For any $n \geq 3$ and any convex body $K$ in the plane, let $C(n,K)$ be the largest quantity such that in every configuration of $n$ points in $K$, there exists a triple of points determining a triangle of area at most $C(n,K)$ times the area of $K$. Establish upper and lower bounds on $C(n,K)$.

Statement quoted from problem 48 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.29 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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