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Borsuk's Conjecture

Borsuk's conjecture (Borsuk 1932) asserts that every bounded set of n-dimensional Euclidean space of generalized diameter 1 can be partitioned into n+1 pieces of strictly smaller generalized diameter. It is true for n=2 and 3 and for sets with smooth boundary. However, the number of pieces required in the worst case grows at least exponentially in sqrt(n), so the conjecture is false in sufficiently 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.

Bondarenko (2014) reduced the dimension of a counterexample to 65, and Jenrich and Brouwer (2014) reduced it to 64. With assistance from GPT-5.5 Pro, Grinsztajn (2026) supplied a construction in dimension 63 with 321 points such that a subset of smaller generalized diameter has at most five points. At least [321/5]=65 pieces are therefore required, where [x] is the ceiling function, exceeding 63+1. This lowers the smallest dimension with a known counterexample to 63 without determining the smallest possible dimension.

See also

Generalized Diameter, Keller's Conjecture, Lebesgue Minimal Problem

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References

Bondarenko, A. "On Borsuk's Conjecture for Two-Distance Sets." Discrete Comput. Geom. 51, 509-515, 2014. https://doi.org/10.1007/s00454-014-9579-4.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.Grinsztajn, M. "A 63-Dimensional Counterexample to Borsuk's Conjecture." 2026. https://github.com/maaxgrin/borsuk-63-counterexample.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 (Ed. V. Klee). 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.Jenrich, T. and Brouwer, A. E. "A 64-Dimensional Counterexample to Borsuk's Conjecture." Electron. J. Combin. 21, #P4.29, 2014. https://doi.org/10.37236/4069.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.

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Borsuk's Conjecture

Cite this as:

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

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