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We consider random lines in $\mathbb{R}^3$ (random with respect to thekinematic measure) and how they intersect $\mathbb{S}^2$. It is known that theentry point and the exit point behave like \textit{independent} uniformlydistributed random variables. We give a new proof using bilinear integralgeometry 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 withthis property (i.e., the intersection points with a random line areindependent), then $n=3$ and $K$ is a ball.

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