Sampling by Intersections with Random Geodesics

TL;DR

This paper compares intersection phenomena of geometric objects with random geodesics on spheres and convex bodies, revealing probabilistic bounds and zero-one laws in high dimensions using Radon transform analysis.

Contribution

It introduces a comparative analysis of intersection behaviors on spheres and convex bodies, employing Radon transform techniques and extending to higher-dimensional subspaces and discrete structures.

Findings

On the sphere, intersection length deviations are bounded with probability away from zero.

In convex bodies, a zero-one law emerges as dimension increases.

Analysis extends to higher-dimensional subspaces and discrete tori.

Abstract

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