A Random Line Intersects S2 in Two Probabilistically Independent Locations

We consider random lines in R3 (random with respect to the kinematic measure) and how they intersect S2. 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⊂Rn 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.