Borsuk’s partition problem

Abstract

If a bounded set Let F ⊂ Rⁿ consists of at least two points, then there exists a finite partition F = F₁ ⋃ ⋯ ⋃ F _k such that for every i = 1, ⋯, k the diameter diam F _i = sup of the part F _i is smaller than diam F. The least positive integer k for which such a partition exists is said to be the Borsuk number of k, since K. Borsuk considered this question for two-dimensional sets and for the n-dimensional ball B ⊂ Rⁿ. One motivation for these investigations was given by the famous theorem of Borsuk and Ulam, referring to continuous mappings of the n-sphere into Rⁿ.