Borsuk's Conjecture

Borsuk conjectured that it is possible to cut an $n$-D shape of 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 $\sim 1.1^{\sqrt{n}}$. Since $1.1^{\sqrt{n}} > n+1$ at $n=9162$, the conjecture becomes false at high dimensions. In fact, the limit has been pushed back to ~ 2000.

See also Diameter (General), Keller's Conjecture, Lebesgue Minimal Problem

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.

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.