Borsuk's conjecture
The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry.
Problem
In 1932 Karol Borsuk showed[1] 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 in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?[1]
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:
- d = 2 — which is the original result by Karol Borsuk (1932).
- d = 3 — shown by Julian Perkal (1947),[2] and independently, 8 years later, by H. G. Eggleston (1955).[3] 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).[4][5]
- For all d for centrally-symmetric bodies — shown by A.S. Riesling (1971).[6]
- For all d for bodies of revolution — shown by Boris Dekster (1995).[7]
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.[8] 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,[9] [10] the current best bound, due to Thomas Jenrich, is 64.[11][12]
Apart from finding the minimum number d 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 d big enough), one needs number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if d is sufficiently large, .[13] The correct order of magnitude of α(d) is still unknown.[14] However, it is conjectured that there is a constant c > 1 such that for all d ≥ 1.
See also
- Hadwiger's conjecture on covering convex bodies with smaller copies of themselves
References
- ^ a b Borsuk, Karol (1933), "Drei Sätze über die n-dimensionale euklidische Sphäre" (PDF), Fundamenta Mathematicae, 20: 177–190
- ^ Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloqium 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), "Borsuk's problem in three-dimensional spaces of constant curvature", Ukr. Geom. Sbornik, 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
- ^ Bondarenko, Andriy V. (2013), On Borsuk’s conjecture for two-distance sets, arXiv:1305.2584
- ^ 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
- ^ Jenrich, Thomas (2013), A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206
- ^ 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
Further reading
- Oleg Pikhurko, Algebraic Methods in Combinatorics, course notes.
- Aicke Hinrichs and Christian Richter, New sets with large Borsuk numbers[permanent dead link], Discrete Mathematics 270 (2003), 137–147
- 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.