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In this paper we compare the different phenomena that occur when intersectinggeometric objects with random geodesics on the unit sphere and inside convexbodies. On the high dimensional sphere we see that with probability boundedaway from zero, the observed length will deviate from the actual measure by atmost a fixed error for any subset, while in convex bodies we can always choosea subset for which the behavior would be close to a zero-one law, as thedimension grows. The result for the sphere is based on an analysis of the Radontransform. Using similar tools we analyze the variance of intersections on thesphere by higher dimensional random subspaces, and on the discrete torus byrandom arithmetic progressions.

Sampling by Intersections with Random Geodesics