A random line intersects ${\mathbb{S}}^2$ in two probabilistically independent locations

Abstract

We consider random lines in ${\mathbb{R}}^3$ (random with respect to the kinematic measure) and how they intersect ${\mathbb{S}}^2$. It is known that the entry point and the exit point behave like \textit{independent} uniformly distributed random variables. We give a new proof using bilinear integral geometry and use this approach to show that this property is extremely rare: if $K\subset {\mathbb{R}}^n$ is a bounded, convex domain with smooth boundary with this property (i.e., the intersection points with a random line are independent), then $n=3$ and $K$ is a ball.