Specifically, I want to determine all distinct “types” of Radon partitions of $ n+2$ points in $ \mathbb{R}^n$ for which the affine hull is all of $ \mathbb{R}^n$ . This is a homework question, so I’m primarily looking for advice on getting started.

Radon’s theorem states that any set of $ n+2$ points in $ \mathbb{R}^n$ can be partitioned into two disjoint sets with intersecting convex hulls.

The affine hull of a set $ S$ is given by $ $ \mbox{aff}(S)=\left\{\sum_{i=1}^k\alpha_ix_i:k>0,x_i\in S,\alpha_i\in\mathbb{R},\sum_{i=1}^k\alpha_i=1\right\}.$ $

So, the simplest case is when $ n=1$ , and clearly the affine hull of any Radon partition of $ n+2=3$ points in $ \mathbb{R}$ is all of $ \mathbb{R}$ . When $ n=2$ , the $ n+2=4$ points can be partitioned as a triple and a singleton or two pairs of points. In the case of the former, the convex hull of the triple must contain the singleton. In the case of the latter, the pairs of points must form intersecting line segments. But the former is the only “type” of partition I want to consider, as it’s affine hull is in fact $ \mathbb{R}^2$ , but the affine hull of the partitions in the latter is not all of $ \mathbb{R}^2$ . So I might conjecture that the full classification requires that the convex hull of one of the partitions must be fully contained in the convex hull of the other partition, but I’m not sure if I even believe this intuitively.

Thanks in advance for any advice.