In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from zero, the observed length will deviate from the actual measure by at most a fixed error for any subset, while in convex bodies we can always choose a subset for which the behavior would be close to a zero-one law, as the dimension grows. The result for the sphere is based on an analysis of the Radon transform. Using similar tools we analyze the variance of intersections on the sphere by higher dimensional random subspaces, and on the discrete torus by random arithmetic progressions.

中文翻译：

---

通过与随机测地线的交叉点进行采样

在本文中，我们比较了几何对象与单位球体上和凸体内部的随机测地线相交时发生的不同现象。在高维球体上，我们看到概率远离零，对于任何子集，观察到的长度将与实际测量偏差至多一个固定误差，而在凸体中，我们总是可以选择一个子集，其行为将是随着维度的增长，接近于零一定律。球体的结果基于对 Radon 变换的分析。使用类似的工具，我们通过高维随机子空间分析球体上交点的方差，通过随机等差数列分析离散环面上的交点方差。