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Level B · Reproducible Geometry P-heilbronn-problem-in-an-arbitrary-convex-bounding-box

Heilbronn problem in an arbitrary convex bounding box

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

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-heilbronn-problem-in-an-arbitrary-convex-bounding-box,
  title        = {Heilbronn problem in an arbitrary convex bounding box},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/heilbronn-problem-in-an-arbitrary-convex-bounding-box}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The problem

For any let be the largest quantity such that in every configuration of points in the plane, there exists a triple of points determining a triangle of area at most times the area of their convex hull. Establish upper and lower bounds on .

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.