Abstract: Given a subset of the sphere and a random subspace, what can be said about the measure of their intersection compared to the measure of the subset?

This question has very different flavour when one considers the high dimensional case, where most methods of study are based on the concentration of measure phenomena, and the low dimensional case where these methods can’t be applied.Our analysis in the low dimensional case focuses on understanding the singular values of Radon type transforms.

We will discuss this question in both scales of dimension and also see that intersections with random geodesics on the two dimensional sphere are very different from intersections with random geodesics inside a convex body.