Remarks on a Generalization of Radon’s Theorem

Abstract

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 ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2$ of $\mathrm{\Delta }$ such that $f({\mathrm{\Delta }}_1)\cap f({\mathrm{\Delta }}_2)\ne \mathrm{\varnothing }$.