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

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. 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
  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), "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
  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:math/9307229, doi:10.1090/S0273-0979-1993-00398-7, MR 1193538
  10. Weißbach, Bernulf (2000), "Sets with Large Borsuk Number" (PDF), Beiträge zur Algebra und Geometrie, 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 V. (2013), On Borsuk's conjecture for two-distance sets, arXiv:1305.2584, Bibcode:2013arXiv1305.2584B
  13. 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
  14. Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206, Bibcode:2013arXiv1308.0206J
  15. Jenrich, Thomas; Brouwer, Andries E. (2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture", Electronic Journal of Combinatorics, 21 (4): #P4.29, MR 3292266
  16. Schramm, Oded (1988), "Illuminating sets of constant width", Mathematika, 35 (2): 180–189, doi:10.1112/S0025579300015175, MR 0986627
  17. 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

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