Unimodality for Radon partitions of random vectors

TL;DR

This paper proves strong unimodality results for the distribution of sizes in Radon partitions of random Gaussian vectors in high-dimensional space, revealing new probabilistic properties of these partitions.

Contribution

It introduces novel unimodality results for the distribution of Radon partition sizes of Gaussian vectors, a previously unexplored aspect.

Findings

Distribution $(p_0,...,p_n)$ is strongly unimodal.

Probabilities $p_k$ exhibit a unimodal pattern.

Results apply to high-dimensional Gaussian vectors.

Abstract

Consider the (almost surely) unique Radon partition of a set of $n$ random Gaussian vectors in $\mathbb R^{n-2}$; choose one of the two parts of this partition uniformly at random, and for $0 \le k \le n$, let $p_k$ denote the probability that it has size $k$. In this paper, we prove strong unimodality results for the distribution $(p_0,\dots,p_n)$.