Borsuk's Conjecture

Borsuk conjectured that it is possible to cut an -D shape of Diameter 1 into pieceseach with diameter smaller than the original. It is true for , 3 and when the boundary is ``smooth.'' However, theminimum number of pieces required has been shown to increase as . Since at, the conjecture becomes false at high dimensions. In fact, the limit has been pushed back to ~ 2000.

References

Borsuk, K. ``Über die Zerlegung einer Euklidischen -dimensionalen Vollkugel in Mengen.'' Verh. Internat. Math.-Kongr. Zürich 2, 192, 1932.

Borsuk, K. ``Drei Sätze über die -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.

Kalai, J. K. G. ``A Counterexample to Borsuk's Conjecture.'' Bull. Amer. Math. Soc. 329, 60-62, 1993.Listernik, L. and Schnirelmann, L. Topological Methods in Variational Problems. Moscow, 1930.

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