KKT Stresses, Affine Moments, and Separator Flux in the Heilbronn Triangle Problem
Abstract
For n points in the unit square, the Heilbronn triangle problem asks for the largest possible minimum triangle area. We develop a variational stress theory for this max-min problem. At every positive-area local optimum, normalized Karush-Kuhn-Tucker multipliers assemble into a skew matrix B satisfying Bz = 2ib, where b is the outward square reaction. This equilibrium has the isotropic affine moment sum_i p_i b_i^T = Delta I_2; the identity also holds for every tight subfamily carrying weights inherited from the same multiplier, with Delta scaled by its multiplier mass. It follows that every positive stress component meets all four sides and that there are at most two such components. To handle nonunique multipliers, we introduce the intersection of the stress kernels over the whole KKT face and a canonical hybrid operator incorporating all tight determinant and boundary derivatives. A strictly convex selector removes every decomposable invisible motion, leaving a rank-one-free residual with a sharp dimension bound; the literal maximal two-dimensional residual is excluded. Two-terminal substresses satisfy an exact interface-flux law, and a five-internal-vertex rank-four block has a Pfaffian cofactor carrier. Finally, an analytic reduction followed by exact symbolic enumeration excludes every one-external completion of a specified one-silent five-cycle residual, for all orientation words. These results isolate the remaining degeneracy but do not solve the problem for arbitrary n.
Related
- Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for n ≤ 8
- The Dirichlet problem for Poisson’s equation in a right-isosceles triangle
- Hypergraph Turán Problem of the Generalized Triangle with Bounded Matching Number
- Averages of Artin characters and their relation to the Davenport--Heilbronn theorem
- A graded version of the separator theorem for planar graphs
BibTeX
@misc{trela2026stresses,
title = {{KKT Stresses, Affine Moments, and Separator Flux in the Heilbronn Triangle Problem}},
author = {Dawid Trela},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25035},
eprint = {2609.25035},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25035},
}
RIS
TY - UNPB
TI - KKT Stresses, Affine Moments, and Separator Flux in the Heilbronn Triangle Problem
AU - Dawid Trela
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25035
UR - https://arxiv.org/abs/2609.25035
AB - For n points in the unit square, the Heilbronn triangle problem asks for the largest possible minimum triangle area. We develop a variational stress theory for this max-min problem. At every positive-area local optimum, normalized Karush-Kuhn-Tucker multipliers assemble into a skew matrix B satisfying Bz = 2ib, where b is the outward square reaction. This equilibrium has the isotropic affine moment sum_i p_i b_i^T = Delta I_2; the identity also holds for every tight subfamily carrying weights inherited from the same multiplier, with Delta scaled by its multiplier mass. It follows that every positive stress component meets all four sides and that there are at most two such components. To handle nonunique multipliers, we introduce the intersection of the stress kernels over the whole KKT face and a canonical hybrid operator incorporating all tight determinant and boundary derivatives. A strictly convex selector removes every decomposable invisible motion, leaving a rank-one-free residual with a sharp dimension bound; the literal maximal two-dimensional residual is excluded. Two-terminal substresses satisfy an exact interface-flux law, and a five-internal-vertex rank-four block has a Pfaffian cofactor carrier. Finally, an analytic reduction followed by exact symbolic enumeration excludes every one-external completion of a specified one-silent five-cycle residual, for all orientation words. These results isolate the remaining degeneracy but do not solve the problem for arbitrary n.
ER -