Radon Partitions of Random Gaussian Polytopes

Abstract

In this paper we study a probabilistic framework for Radon partitions, where our points are chosen independently from the $d$-dimensional normal distribution. For every point set we define a corresponding Radon polytope, which encodes all information about Radon partitions of our set - with Radon partitions corresponding to faces of the polytope. This allows us to derive expressions for the probability that a given partition of $N$ randomly chosen points in $\mathbb{R}^d$ forms a Radon partition. These expressions involve conic kinematic formulas and intrinsic volumes, and in general require repeated integration, though we obtain closed formulas in some cases. This framework can provide new perspectives on open problems that can be formulated in terms of Radon partitions, such as Reay's relaxed Tverberg conjecture.