Английская Википедия:Borsuk's conjecture
The Borsuk problem in geometry, for historical reasonsШаблон:Refn incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk.
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 Шаблон:Mvar-dimensional ball can be covered with Шаблон:Math compact sets of diameters smaller than the ball. At the same time he proved that Шаблон:Mvar subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:[1]
The question was answered in the positive in the following cases:
- Шаблон:Math — which is the original result by Karol Borsuk (1932).
- Шаблон:Math — 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 Шаблон:Mvar for smooth convex bodies — shown by Hugo Hadwiger (1946).[4][5]
- For all Шаблон:Mvar for centrally-symmetric bodies — shown by A.S. Riesling (1971).[6]
- For all Шаблон:Mvar 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 Шаблон:Em.[8] They claim that their construction shows that Шаблон:Math pieces do not suffice for Шаблон:Math and for each Шаблон:Math. However, as pointed out by Bernulf Weißbach,[9] 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 Шаблон:Math (as well as all higher dimensions up to 1560).[10]
Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for Шаблон:Math, which cannot be partitioned into Шаблон:Math parts of smaller diameter.[11]
In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all Шаблон:Math.[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 Шаблон:Mvar of dimensions such that the number of pieces Шаблон:Math, mathematicians are interested in finding the general behavior of the function Шаблон:Math. Kahn and Kalai show that in general (that is, for Шаблон:Mvar sufficiently large), one needs <math display="inline">\alpha(n) \ge (1.2)^\sqrt{n}</math> many pieces. They also quote the upper bound by Oded Schramm, who showed that for every Шаблон:Mvar, if Шаблон:Mvar is sufficiently large, <math display="inline">\alpha(n) \le \left(\sqrt{3/2} + \varepsilon\right)^n</math>.[15] The correct order of magnitude of Шаблон:Math is still unknown.[16] However, it is conjectured that there is a constant Шаблон:Math such that Шаблон:Math for all Шаблон:Math.
See also
- Hadwiger's conjecture on covering convex bodies with smaller copies of themselves
- Kahn–Kalai conjecture
Note
References
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.
- Шаблон:Cite book
External links
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