Borsuk conjectured that it is possible to cut an -D shape of 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. 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 -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.