# Borsuk's conjecture

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An example of a hexagon cut into three pieces of smaller diameter.

The Borsuk problem in geometry, for historical reasons[note 1] incorrectly called Borsuk's conjecture, is a question in discrete geometry.

## Problem

In 1932 Karol Borsuk showed[2] that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally d-dimensional ball can be covered with d + 1 compact sets of diameters smaller than the ball. At the same time he proved that d subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:

Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes ${\displaystyle {\mathbb {R}}^{n}}$ in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?[2]

This can be translated as:

The following question remains open: Can every bounded subset E of the space ${\displaystyle {\mathbb {R}}^{n}}$ be partitioned into (n + 1) sets, each of which has a smaller diameter than E?

The question got a positive answer in the following cases:

• d = 2 — which is the original result by Karol Borsuk (1932).
• d = 3 — shown by Julian Perkal (1947),[3] and independently, 8 years later, by H. G. Eggleston (1955).[4] A simple proof was found later by Branko Grünbaum and Aladár Heppes.
• For all d for the smooth convex bodies — shown by Hugo Hadwiger (1946).[5][6]
• For all d for centrally-symmetric bodies — shown by A.S. Riesling (1971).[7]
• For all d for bodies of revolution — shown by Boris Dekster (1995).[8]

The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is no.[9] Their construction shows that d + 1 pieces do not suffice for d = 1,325 and for each d > 2,014.

After Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all d ≥ 65,[10] [11] the current best bound, due to Thomas Jenrich, is 64.[12][13]

Apart from finding the minimum number d of dimensions such that the number of pieces ${\displaystyle \alpha (d)>d+1}$ mathematicians are interested in finding the general behavior of the function ${\displaystyle \alpha (d)}$. Kahn and Kalai show that in general (that is for d big enough), one needs ${\displaystyle \alpha (d)\geq (1.2)^{\sqrt {d}}}$ number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if d is sufficiently large, ${\displaystyle \alpha (d)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{d}}$.[14] The correct order of magnitude of α(d) is still unknown.[15] However, it is conjectured that there is a constant c > 1 such that ${\displaystyle \alpha (d)>c^{d}}$ for all d ≥ 1.

## Note

1. ^ As Hinrichs and Richter say in the introduction to their work[1], the “Borsuk's conjecture [was] believed by many to be true for some decades” (hence commonly called 'a conjecture') so “it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary”. It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.

## References

1. ^ Hinrichs, Aicke; Richter, Christian (28 August 2003). "New sets with large Borsuk numbers". Discrete Mathematics. Elsevier. 270 (1–3): 137–147. doi:10.1016/S0012-365X(02)00833-6.
2. ^ a b Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" (PDF), Fundamenta Mathematicae (in German), 20: 177–190
3. ^ Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloquium Mathematicum, 2: 45
4. ^ Eggleston, H. G. (1955), "Covering a three-dimensional set with sets of smaller diameter", Journal of the London Mathematical Society, 30: 11–24, doi:10.1112/jlms/s1-30.1.11, MR 0067473
5. ^ Hadwiger, Hugo (1945), "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici, 18 (1): 73–75, doi:10.1007/BF02568103, MR 0013901
6. ^ Hadwiger, Hugo (1946), "Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici, 19 (1): 72–73, doi:10.1007/BF02565947, MR 0017515
7. ^ Riesling, A. S. (1971), "Borsuk's problem in three-dimensional spaces of constant curvature" (PDF), Ukr. Geom. Sbornik (in Russian), Kharkov State University (now National University of Kharkiv), 11: 78–83
8. ^ Dekster, Boris (1995), "The Borsuk conjecture holds for bodies of revolution", Journal of Geometry, 52 (1-2): 64–73, doi:10.1007/BF01406827, MR 1317256
9. ^ Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society, 29 (1): 60–62, arXiv:, doi:10.1090/S0273-0979-1993-00398-7, MR 1193538
10. ^ Bondarenko, Andriy V. (2013), On Borsuk’s conjecture for two-distance sets, arXiv:
11. ^ Bondarenko, Andriy (2014), "On Borsuk's Conjecture for Two-Distance Sets", Discrete & Computational Geometry, 51 (3): 509–515, doi:10.1007/s00454-014-9579-4, MR 3201240
12. ^ Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:
13. ^ Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture", Electronic Journal of Combinatorics, 21 (4): #P4.29, MR 3292266
14. ^ Schramm, Oded (1988), "Illuminating sets of constant width", Mathematika, 35 (2): 180–189, doi:10.1112/S0025579300015175, MR 0986627
15. ^ Alon, Noga (2002), "Discrete mathematics: methods and challenges", Proceedings of the International Congress of Mathematicians, Beijing, 1: 119–135, arXiv: