Heilbronn minimum tetrahedron volume
Place n points in the unit cube so that the smallest tetrahedron formed by any four of them is as large as possible.
Formal definition
- ContainerThe unit cube [0, 1]³ with opposite corners at (0, 0, 0) and (1, 1, 1)
- SubmissionExactly n points, no four of them coplanar
- ConstraintsEvery point lies inside the cube or on its boundary
- ObjectiveMaximize the minimum tetrahedron volume over all quadruples, compared exactly via the 3×3 determinant of three edge vectors
Getting a feel for it
Coplanar and collinear degeneracies
In the plane, three points degenerate only when they are collinear; in three dimensions, four points degenerate whenever they are coplanar or three of them are collinear. Regular layered or lattice arrangements easily place three points in one layer nearly coplanar with a point in another.
Known results and open cases
While the planar Heilbronn triangle problem is proved optimal for n = 5–9 and has extensive numerical tables beyond that, the three-dimensional tetrahedron problem remains open for all n ≥ 6, with only n = 5 proved optimal at volume 1/6.
Current best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (0) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/heilbronn-tetrahedron.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/heilbronn-tetrahedron.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-tetrahedron-2026-10,
title = {{Heilbronn minimum tetrahedron volume} (P72)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/heilbronn-tetrahedron.json},
license = {CC BY 4.0}
}Discussion
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