Dimension of the Radon set

Abstract

We consider when a subset \(X\subset\mathbb{R}^{d}\) has a Radon partition \(X=X_{1}\sqcup X_{2}\) such that

$$\dim(({\rm conv} X_{1})\cap({\rm conv} X_{2}) )= \min\lbrace \dim({\rm conv} X_{1}), \dim({\rm conv} X_{2})\rbrace, $$

showing that such a partition always exists when \(X\) has at least \(\lfloor\frac{3d}{2}\rfloor+2\) points in general position. The latter bound is sharp.