Borsuk's conjecture

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

Unsolved problem in mathematics

What is the lowest n such that not every bounded subset E of the space ${\displaystyle \mathbb {R} ^{n}}$ can be partitioned into (n + 1) sets, each of which has a smaller diameter than E?

More unsolved problems in mathematics

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: ^[2]

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? 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?

— Drei Sätze über die n-dimensionale euklidische Sphäre

The question was answered in the positive 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 fields — shown by Hugo Hadwiger (1946). ^{[5] [6]}
• For all n for centrally-symmetric fields — shown by A.S. Riesling (1971). ^[7]
• For all n for fields 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] Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now. ^{[13] [14]}

Apart from finding the minimum number n of dimensions such that the number of pieces α(n) > n + 1, mathematicians are interested in finding the general behavior of the function α(n). Kahn and Kalai show that in general (that is, for n sufficiently large), one needs ${\textstyle \alpha (n)\geq (1.2)^{\sqrt {n}}}$ many pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if n is sufficiently large, ${\textstyle \alpha (n)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{n}}$. ^[15] The correct order of magnitude of α(n) is still unknown. ^[16] However, it is conjectured that there is a constant c > 1 such that α(n) > cⁿ for all n ≥ 1.

Oded Schramm also worked on a related question, a body ${\displaystyle K}$ of constant width is said to have effective radius ${\displaystyle r}$ if ${\displaystyle {\text{Vol}}(K)=r^{n}{\text{Vol}}(\mathbb {B} ^{n})}$, where ${\displaystyle \mathbb {B} ^{n}}$ is the unit ball in ${\displaystyle \mathbb {R} ^{n}}$, he proved the lower bound ${\displaystyle {\sqrt {3+2/(n+1)}}-1\leq r_{n}}$, where ${\displaystyle r_{n}}$ is the smallest effective radius of a body of constant width 2 in ${\displaystyle \mathbb {R} ^{n}}$ and asked if there exists ${\displaystyle \epsilon >0}$ such that ${\displaystyle r_{n}\leq 1-\epsilon }$ for all ${\displaystyle n\geq 2}$, ^{[17] [18]} that is if the gap between the volumes of the smallest and largest constant-width bodies grows exponentially. In 2024 a preprint by Arman, Bondarenko, Nazarov, Prymak, Radchenko reported to have answered this question in the affirmative giving a construction that satisfies ${\displaystyle {\text{Vol}}(K)\leq (0.9)^{n}{\text{Vol}}(\mathbb {B} ^{n})}$. ^{[19] [20] [21]}

See also

• Hadwiger's conjecture on covering convex fields with smaller copies of themselves
• Kahn–Kalai conjecture

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". However, 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 . 270 (1–3). Elsevier: 137–147. doi: 10.1016/S0012-365X(02)00833-6 .
2. Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" [Three theorems about the n-dimensional Euclidean sphere](PDF), Fundamenta Mathematicae (in German), 20: 177–190, doi: 10.4064/fm-20-1-177-190
3. Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloquium Mathematicum (in French), 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 (in German), 18 (1): 73–75, doi:10.1007/BF02568103, MR   0013901, S2CID   122199549
6. Hadwiger, Hugo (1946), "Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici (in German), 19 (1): 72–73, doi:10.1007/BF02565947, MR   0017515, S2CID   121053805
7. 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), 11, Kharkov State University (now National University of Kharkiv): 78–83
8. Dekster, Boris (1995), "The Borsuk conjecture holds for fields of revolution", Journal of Geometry, 52 (1–2): 64–73, doi:10.1007/BF01406827, MR   1317256, S2CID   121586146
9. 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, S2CID   119647518
10. Weißbach, Bernulf (2000), "Sets with Large Borsuk Number" (PDF), Beiträge zur Algebra und Geometrie (in German), 41 (2): 417–423
11. Jenrich, Thomas (2018), On the counterexamples to Borsuk's conjecture by Kahn and Kalai, arXiv: 1809.09612v4
12. Bondarenko, Andriy (2014) [2013], "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
13. Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv: 1308.0206 , Bibcode:2013arXiv1308.0206J
14. Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture", Electronic Journal of Combinatorics , 21 (4): #P4.29, doi: 10.37236/4069 , MR   3292266
15. Schramm, Oded (1988), "Illuminating sets of constant width", Mathematika , 35 (2): 180–189, doi:10.1112/S0025579300015175, MR   0986627
16. 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
17. Schramm, Oded (June 1988). "On the volume of sets having constant width" . Israel Journal of Mathematics. 63 (2): 178–182. doi:10.1007/BF02765037. ISSN   0021-2172.
18. Kalai, Gil (2015-05-19). "Some old and new problems in combinatorial geometry I: Around Borsuk's problem". arXiv: 1505.04952 [math.CO].
19. Arman, Andrii; Bondarenko, Andriy; Nazarov, Fedor; Prymak, Andriy; Radchenko, Danylo (2024-05-28). "Small volume bodies of constant width". arXiv: 2405.18501 [math.MG].
20. Kalai, Gil (2024-05-31). "Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko Constructed Small Volume Bodies of Constant Width". Combinatorics and more. Retrieved 2024-09-28.
21. Barber, Gregory (2024-09-20). "Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem". Quanta Magazine. Retrieved 2024-09-28.

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. Vol. 347. Cambridge University Press. pp. 202–247. ISBN   978-0-521-70564-6. Zbl   1144.52005.

External links

• Weisstein, Eric W. "Borsuk's Conjecture". MathWorld .