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quarta-feira, 6 de agosto de 2014

Borsuk's Conjecture Borsuk's problem

Borsuk conjectured that it is possible to cut an n-dimensional shape of generalized diameter 1 into n+1 pieces each with diameter smaller than the original. It is true for n=2, 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as ∼1.1^(sqrt(n)). Since 1.1^(sqrt(n))>n+1 at n=9162, 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 n>297.
  Title: Borsuk's problem
     Author: Raigorodskii AM
     Format: PDF
     Size: 1.05 MB
     Year of Publication: 2006  

LINK
https://www.mediafire.com/?9v4892uf4w4q7ut

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