Borsuk's Conjecture Borsuk's problem

Borsuk conjectured that it is possible to cut an -dimensional shape of generalized diameter 1 into pieces each with diameter smaller than the original. It is true for , 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as . Since at , the conjecture becomes false at high dimensions.
Kahn and Kalai (1993) found a counterexample in dimension 1326, Nilli (1994) a counterexample in dimension 946. Hinrichs and Richter (2003) showed that the conjecture is false for all .
  Title: Borsuk's problem
     Author: Raigorodskii AM
     Format: PDF
     Size: 1.05 MB
     Year of Publication: 2006  
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