Automated Proofs in Geometry : Computing Upper Bounds for the Heilbronn Problem for Triangles

Heilbronn型三角形的平均面积

The Average-Case Area of Heilbronn-Type Triangles

在具有从单位正方形中的$n$点选择的顶点的$ {n \choose 3}$三角形中，令$T$为具有最小面积的三角形，并且令$A$为$T$的面积。海尔布隆三角形问题要求$A$在所有$n$点的选择上假设的最大值。我们考虑平均情况：如果$n$点是独立和随机选择的（具有均匀分布），则存在正常数$c$和$C$，使得对于所有足够大的$n$值，$c/n^3 < μ_n < C/n^3$，其中$μ_n$是$A$的期望。此外，$c/n^3 < A < C/n^3$，概率接近1。我们的证明使用基于Kolmogorov复杂性的不可压缩性方法;它实际上确定了在一般位置上排列的最小三角形的面积。''

使用全局优化方法解决海因布伦三角问题

Solving the Heilbronn Triangle Problem using Global Optimization Methods

本文提出使用全局优化方法解决海因布伦三角问题，通过改进的公式和计算方法，实现了n=9点的最优解，计算效率显著提升。

Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $Ω(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. F…

k边形Heilbronn问题的改进上界

Improved upper bounds for the Heilbronn's Problem for $k$-gons

本研究针对k点凸包版本的Heilbronn问题，解决了Szemerédi提出的改进平凡上界常数的问题，证明了2/n+o(1/n)的上界，并将结果推广到k≥4的任意凸包情形。

Finding all Maximal Area Parallelograms in a Convex Polygon

Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant of computing the maximum area parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs) in a given convex polygon. By proving and utilizing several structural properties of the LMAPs, we compute all of them (including all the MAPs) in $O(n^2)$ time, where $n$ denotes the number of edges of the given polygon. In addition, we prove that the LMAPs interleave each other and thus the number of LMAPs is $O(n)$. We discuss applications of our result to, among …

On Automating Triangle Constructions in Absolute and Hyperbolic Geometry

We describe first steps towards a system for automated triangle constructions in absolute and hyperbolic geometry. We discuss key differences between constructions in Euclidean, absolute and hyperbolic geometry, compile a list of primitive constructions and lemmas used for constructions in absolute and hyperbolic geometry, build an automated system for solving construction problems and test it on a corpus of triangle-construction problems. We also provide an online compendium containing construction descriptions and illustrations.