# Borsuk's conjecture

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

## 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 *n*-dimensional ball can be covered with *n* + 1 compact sets of diameters smaller than the ball. At the same time he proved that *n* 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 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 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:

*n*= 2 — which is the original result by Karol Borsuk (1932).*n*= 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
*n*for smooth convex bodies — shown by Hugo Hadwiger (1946).[5][6] - For all
*n*for centrally-symmetric bodies — shown by A.S. Riesling (1971).[7] - For all
*n*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] They claim that their construction shows that *n* + 1 pieces do not suffice for *n* = 1325 and for each *n* > 2014. However, as pointed out by Bernulf Weißbach,[10] the first part of this claim is in fact false. But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for *n* = 1325 (as well as all higher dimensions up to 1560).[11]

Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for *n* ≥ 298, which cannot be partitioned into *n* + 11 parts of smaller diameter.[1]

In 2013, Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all *n* ≥ 65.[12][13] Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.[14][15]

Apart from finding the minimum number *n* of dimensions such that the number of pieces mathematicians are interested in finding the general behavior of the function . Kahn and Kalai show that in general (that is for *n* big enough), one needs number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every *ε*, if *n* is sufficiently large, .[16] The correct order of magnitude of *α*(*n*) is still unknown.[17] However, it is conjectured that there is a constant *c* > 1 such that for all *n* ≥ 1.

## See also

- Hadwiger's conjecture on covering convex bodies with smaller copies of themselves

## Note

- 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

- 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. - Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" (PDF),
*Fundamenta Mathematicae*(in German),**20**: 177–190, doi:10.4064/fm-20-1-177-190 - Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur",
*Colloquium Mathematicum*,**2**: 45 - 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 - Hadwiger, Hugo (1945), "Überdeckung einer Menge durch Mengen kleineren Durchmessers",
*Commentarii Mathematici Helvetici*,**18**(1): 73–75, doi:10.1007/BF02568103, MR 0013901 - 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 - Riesling, A. S. (1971), "Проблема Борсука в трехмерных пространствах постоянной кривизны" [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 - 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 - Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture",
*Bulletin of the American Mathematical Society*,**29**(1): 60–62, arXiv:math/9307229, doi:10.1090/S0273-0979-1993-00398-7, MR 1193538 - Weißbach, Bernulf (2000), "Sets with Large Borsuk Number" (PDF),
*Beiträge zur Algebra und Geometrie*,**41**(2): 417–423 - Jenrich, Thomas (2018),
*On the counterexamples to Borsuk's conjecture by Kahn and Kalai*, arXiv:1809.09612v4 - Bondarenko, Andriy V. (2013),
*On Borsuk's conjecture for two-distance sets*, arXiv:1305.2584, Bibcode:2013arXiv1305.2584B -
Bondarenko, Andriy (2014), "On Borsuk's Conjecture for Two-Distance Sets",
*Discrete & Computational Geometry*,**51**(3): 509–515, arXiv:1305.2584, doi:10.1007/s00454-014-9579-4, MR 3201240 - Jenrich, Thomas (2013),
*A 64-dimensional two-distance counterexample to Borsuk's conjecture*, arXiv:1308.0206, Bibcode:2013arXiv1308.0206J - Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture",
*Electronic Journal of Combinatorics*,**21**(4): #P4.29, MR 3292266 - Schramm, Oded (1988), "Illuminating sets of constant width",
*Mathematika*,**35**(2): 180–189, doi:10.1112/S0025579300015175, MR 0986627 - Alon, Noga (2002), "Discrete mathematics: methods and challenges",
*Proceedings of the International Congress of Mathematicians, Beijing*,**1**: 119–135, arXiv:math/0212390, Bibcode:2002math.....12390A

## Further reading

- Oleg Pikhurko,
*Algebraic Methods in Combinatorics*, course notes. - Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary,
*Mathematical Intelligencer***26**(2004), no. 3, 4–12. - Raigorodskii, Andreii M. (2008). "Three lectures on the Borsuk partition problem". In Young, Nicholas; Choi, Yemon (eds.).
*Surveys in contemporary mathematics*. London Mathematical Society Lecture Note Series.**347**. Cambridge University Press. pp. 202–247. ISBN 978-0-521-70564-6. Zbl 1144.52005.