AI 中文总结
该研究针对三角形中独立同分布点，得出最长凸链长度主阶为cn^(1/3)，并建立相关变分公式，还与正方形中最长单调链的渐近结果存在关联。
AI 中文摘要
从三角形中按某有界密度函数抽取n个独立同分布（i.i.d.）点，给定三角形的两个顶点A、B，问有多少个样本能形成以A为起点、B为终点的凸链的最大数量？我们证明，在主阶近似下，答案为cn^(1/3，其中c为常数)，推广了Ambrus和Bárány针对均匀分布点的结果。此外，我们用变分公式表示常数c，若该变分公式的最大化元唯一，则其给出最长凸链的极限曲线。作为对比，从单位正方形中抽取n个独立同分布样本时，最长单调链的长度渐近为c'n^(1/2)，尽管尺度不同，我们的公式与Deuschel和Zeitouni建立的c'的公式存在良好关联。
英文摘要
Sample $n$ i.i.d. points from a triangle, according to some bounded density function. Given two vertices $A,B$ of the triangle, what is the maximum number of samples that form a convex chain with initial point $A$ and terminal point $B$? We show that to leading order, the answer is $cn^{1/3}$, generalizing a result of Ambrus and Bárány that considered uniformly distributed points. Furthermore, we express the constant $c$ using a variational formula whose maximizer (if unique) gives the limiting curve formed by the longest convex chain. By comparison, for $n$ i.i.d. samples from the unit square, the length of the longest monotone chain is asymptotically $c'n^{1/2}$. Despite the difference in scale, our formula is nicely connected to one established for $c'$ by Deuschel and Zeitouni.
Comments79 pages, 9 figures
