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Borsuk's Conjecture


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.


See also

Generalized Diameter, Keller's Conjecture, Lebesgue Minimal Problem

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References

Borsuk, K. "Über die Zerlegung einer Euklidischen n-dimensionalen Vollkugel in n Mengen." Verh. Internat. Math.-Kongr. Zürich 2, 192, 1932.Borsuk, K. "Drei Sätze über die n-dimensionale euklidische Sphäre." Fund. Math. 20, 177-190, 1933.Cipra, B. "If You Can't See It, Don't Believe It...." Science 259, 26-27, 1993.Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 21-25, 1993.Grünbaum, B. "Borsuk's Problem and Related Questions." In Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, Held at the University of Washington, Seattle, June 13-15, 1961. Providence, RI: Amer. Math. Soc., pp. 271-284, 1963.Hinrichs, A. and Richter, C. "New Sets with Large Borsuk Numbers." Disc. Math. 270, 137-147, 2003.Kahn, J. and Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture." Bull. Amer. Math. Soc. 29, 60-62, 1993.Lyusternik, L. and Schnirel'mann, L. Topological Methods in Variational Problems. Moscow, 1930.Lyusternik, L. and Schnirel'mann, L. "Topological Methods in Variational Problems and Their Application to the Differential Geometry of Surfaces." Uspehi Matem. Nauk (N.S.) 2, 166-217, 1947.Nilli, A. "On Borsuk's Problem." Jerusalem Combinatorics '93. Papers from the International Conference on Combinatorics held in Jerusalem, May 9-17, 1993 (Ed. H. Barcelo and G. Kalai.) Providence, RI: Amer. Math. Soc., pp. 209-210, 1994.

Cite this as:

Weisstein, Eric W. "Borsuk's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BorsuksConjecture.html

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