Unimodality for Radon partitions of random vectors

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 $(p_0,\dots ,p_n)$.