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

TL;DR

This paper proves that in three-dimensional space, only a spherical convex domain exhibits the property that a random line intersects it in two independent points, highlighting the uniqueness of the sphere.

Contribution

The paper provides a new proof using bilinear integral geometry and characterizes the unique convex domain with independent intersection points as the sphere in three dimensions.

Findings

Only in 3D does the independence property hold for convex domains.

The sphere is uniquely characterized by this property among smooth convex bodies.

The approach uses bilinear integral geometry to establish the result.

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.