论文标题
扁平托里的随机三角形
Random triangles on flat tori
论文作者
论文摘要
受到询问几何现象概率的几何形状中的经典难题的启发，我们给出了一个明确的公式，即在平坦的圆环上随机三角形是同型琐碎的概率。我们进行此计算的主要工具涉及将问题降低为在平面上不变的平面中可测量集合的新不变式，该集合在区域保护仿射变换下没有变化。我们的结果表明，这种概率在所有矩形托里最小化，并在常规的六边形圆环中最大化。
Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.
