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 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.
