发表机构
Doctoral School and Faculty of Law and Administration, War Studies University（华沙战争大学法律与管理学院博士学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Heilbronn三角形问题，提出变分应力理论，通过KKT乘子、仿射矩和通量定律刻画最优解结构，并排除部分退化情形，但未完全解决任意n的问题。
AI 中文摘要
对于单位正方形中的n个点，Heilbronn三角形问题要求最大化最小三角形面积。我们为该最大最小问题发展了一种变分应力理论。在每个正面积局部最优解处，归一化的Karush-Kuhn-Tucker乘子组装成一个斜对称矩阵B，满足Bz = 2ib，其中b是向外的正方形反作用力。该平衡具有各向同性的仿射矩和sum_i p_i b_i^T = Delta I_2；该恒等式也适用于每个携带来自同一乘子继承权重的紧子族，其中Delta按乘子质量缩放。由此可知，每个正应力分量都与所有四条边相交，且至多存在两个这样的分量。为处理非唯一乘子，我们引入了整个KKT面上应力核的交集，以及一个包含所有紧行列式和边界导数的典型混合算子。一个严格凸的选择器移除所有可分解的不可见运动，留下一个无秩一的残差，并具有尖锐的维数界；字面意义上的最大二维残差被排除。两端子子应力满足精确的界面通量定律，一个五内顶点秩四块具有Pfaffian余因子载体。最后，解析化简后接精确符号枚举，对所有方向词排除了指定的一静五环残差的每一个单外顶点补全。这些结果隔离了剩余的退化性，但并未解决任意n的问题。
英文摘要
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.
Comments37 pages. Includes exact symbolic verification of the finite residual cases and reproducibility material
