Borsuk's partition conjecture
This is a survey article on the famous problem of \textit{K. Borsuk} [Fundam. Math. 20, 177--190 (1933; Zbl 0006.42403)] who asked if every set of a given diameter (we tacitly assume that over 0) in the Euclidean \(n\)-dimensional space \(E^n\) can be divided into \(n+1\) parts of smaller diameter. First, this article lists some cases when the problem has a positive solution. For instance for \(n=3\) and for every convex body with smooth boundary in \(E^n\). Then much attention is paid to the result by \textit{J. Kahn} and \textit{G. Kalai} [Bull. Am. Math. Soc., New Ser. 29, No. 1, 60--62 (1993; Zbl 0786.52002)] who showed that the problem has a negative solution for sufficiently large \(n\). Next, the author recalls a number of estimates on the number of these parts. He also considers a related covering problem by \textit{H. Hadwiger} [``Ungelöste Probleme. 20'', Elem. Math. 12, 121 (1957)]. The last section of this article is entitled ``Computer program for Borsuk's conjecture''. It proposes to apply a net of lattice polytopes and the Hausdorff distances to any given set that is expected to be partitioned. The author adds that even for \(n=4\) ``it's to hard for todays computers'' since the unknown number of such lattice polytopes is at most \(e^{(\omega_4 \cdot 2760^4)^{3/5}}\), where \(\omega_4\) denotes the volume of the unit ball of \(E^4\).
scientific article; zbMATH DE number 7458828
scientific article