Radon Partitions of Random Gaussian Polytopes

TL;DR

This paper introduces a probabilistic framework for Radon partitions of random Gaussian point sets, using Radon polytopes to derive probability expressions and explore connections to open geometric problems.

Contribution

It develops a novel polytope-based approach to analyze Radon partitions in Gaussian random point sets, providing explicit probability formulas and insights into related conjectures.

Findings

Derived probability expressions involving conic kinematic formulas

Obtained closed-form formulas in specific cases

Provided new perspectives on open problems like Reay's Tverberg conjecture

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.