Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for n ≤ 8

Abstract

We study Heilbronn's triangle problem in the unit right triangle, where n points are placed to maximize the smallest of the binomn3 triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with n ≥ 5 has at least four points on the boundary, one edge containing two of them. With the affine S₃ symmetry this fixes four boundary points and n orientation variables in a mixed-integer model that certifies global optimality for all n ≤ 8 up to a relative gap of 10⁻⁴. For n ≤ 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 S₇, so on that conjecture no expression in radicals exists.

KKT Stresses, Affine Moments, and Separator Flux in the Heilbronn Triangle Problem
Dawid Trela
arXiv1 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics Geometry & topologyarXiv
When Gigawatts of Computational Load Disappear: Cycle-Space Certificates for Grid Synchronization and Transient Stability
Michael Chertkov
arXiv23 Sep 2026PreprintarXiv licenceFull textDynamical systems Statistical & nonlinear physicsarXiv
Binary Minor Certificates for the 𝒜(1)-hit problem of MO(2)
Dang Vo Phuc
arXiv8 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Proving the optimality of the product construction for the intersection--union problem
Fan Chang, Peter Frankl, Jian Wang
arXiv18 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
The Dirichlet problem for Poisson’s equation in a right-isosceles triangle
Juliana Zago Dias Feijó, Vinícius Coutinho da Silva, André von Borries Lopes
Boundary Value Problems25 Oct 2026Peer-reviewedCC BYFull textGeneral mathematicsDOAJ

Cite

BibTeX RIS

BibTeX

@misc{sudermannmerx2026heilbronns,
  title = {{Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for $n \le 8$}},
  author = {Nathan Sudermann-Merx},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.15021},
  eprint = {2607.15021},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.15021},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for n ≤ 8
AU  - Nathan Sudermann-Merx
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.15021
UR  - https://arxiv.org/abs/2607.15021
AB  - We study Heilbronn's triangle problem in the unit right triangle, where n points are placed to maximize the smallest of the binomn3 triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with n ≥ 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 ≤ 8 up to a relative gap of 10^−4. For n ≤ 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.
ER  -

Abstract

Related

Cite