Borsuk's partition problem and finite point sets
In his famous paper [Fundam. Math. 20, 177--190 (1933; Zbl 0006.42403)], \textit{K. Borsuk} asked if every bounded subset \(X\) of \(\mathbb{R}^{n}\) can be decomposed into \(n+1\) subsets with diameters smaller than \(\operatorname{diam} X\). In this paper, the author shows that this is equivalent to a similiar question on finite subsets of \(\mathbb{R}^{n}\) or of \(\mathbb{Z}^{n}\) by a general result of the following theorem on coverings of compact metric spaces. Let \(Y\) be a compact metric space, \(k\in\mathbb{N}\) and \(t\in\mathbb{R}\). Then the following statements are equivalent: {\parindent=0.7cm\begin{itemize}\item[(i)] Every subset \(X \subseteq Y\) with \(\operatorname{diam}X=1 \) is the union of \( k \) subsets with diameters smaller than \( t \). \item [(ii)] Every countable subset \(X \subseteq Y\) with \( \operatorname{diam} X=1 \) is the union of \( k \) subsets with diameters smaller than \( t \). \item [(iii)] For some real number \(c<t\), every subset \(X \subseteq Y\) with \(\operatorname{diam} X=1 \) is the union of \( k \) subsets with diameters at most \( c \). \item[(iv)] For some real number \(c<t\), every finite subset \(X \subseteq Y\) with \(\operatorname{diam} X=1 \) is the union of \( k \) subsets with diameters at most \( c \). \end{itemize}} If these conditions hold, then one can take the same constant \( c \) in (iii) and (iv). Moreover, the author also considers Borsuk's problem in \(\mathbb{Q}^{4}\).
scientific article; zbMATH DE number 6971510
scientific article