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### Borsuk's Conjecture

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 .

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