Remarks on a Generalization of Radon’s Theorem

Let \(S^n\) be the \(n\)-dimensional sphere and \(K\) be the simplicial complex consisting of all faces of some \((n+1)\)-dimensional simplex. We present an explicit construction of a function \(g: S^n \to |K|\) such that for every \(x \in S^n\), the supports of \(g(x)\) and \(g(-x)\) are disjoint. This construction provides a new proof of the following result of Bajméczy and Bérdny \([1]\), which is a generalization of a theorem of Radon \([4]\):
If \(f: |K| \to \mathbb{R}^n\) is a continuous map, then there are two disjoint faces \(\Delta_1, \Delta_2\) of \(\Delta\) such that \(f(\Delta_1) \cap f(\Delta_2) \neq \emptyset\).