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| [[File:Borsuk Hexagon.svg|200px|thumb|right|An example of a [[hexagon]] cut into three pieces of smaller diameter.]]
| | The name of the writer is Luther. Meter reading is exactly where my main earnings comes from but quickly I'll be on my own. Delaware is our birth location. What she loves performing is to perform croquet but she hasn't made a dime with it.<br><br>Have a look at my weblog :: extended car warranty ([http://Www.carelion.com/UserProfile/tabid/61/userId/107768/Default.aspx description here]) |
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| The '''Borsuk problem in geometry''', for historical reasons incorrectly called '''Borsuk's [[conjecture]]''', is a question in [[discrete geometry]]. | |
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| ==Problem==
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| In 1932 [[Karol Borsuk]] showed<ref name="BorsukFM">K. Borsuk, ''Drei Sätze über die n-dimensionale euklidische Sphäre'', "Fundamenta Mathematicae", '''20''' (1933). 177–190</ref> that an ordinary 3-dimensional [[ball (mathematics)|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 {{nobr|''d'' + 1}} [[Compact space|compact]] [[Set (mathematics)|sets]] of diameters smaller than the ball. At the same time he proved that ''d'' [[subset]]s are not enough in general. The proof is based on the [[Borsuk–Ulam theorem]]. That led Borsuk to a general question:
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| : ''Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes <math>\Bbb R^n</math> in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?''<ref name="BorsukFM" />
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| Translation:
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| : ''The following question remains open: Can every [[bounded set|bounded]] subset E of the space <math>\Bbb R^n</math> be [[partition of a set|partitioned]] into (n + 1) sets, each of which has a smaller diameter than E?''
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| The question got a positive answer in the following cases:
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| * ''d'' = 2 — the original result by Borsuk (1932).
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| * ''d'' = 3 — the result of Julian Perkal (1947),<ref>J. Perkal, Sur la subdivision des ensembles en parties de diamètre inférieur, ''Colloq. Math.'' '''2''' (1947), 45.</ref> and independently, 8 years later, H. G. Eggleston (1955).<ref>H. G. Eggleston, Covering a three-dimensional set with sets of smaller diameter, ''J. Lond. Math. Soc''. 30 (1955), 11–24.</ref> A simple proof was found later by [[Branko Grünbaum]] and Aladár Heppes.
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| * For all ''d'' for the [[Smooth manifold|smooth]] convex bodies — the result of [[Hugo Hadwiger]] (1946).<ref>Hadwiger H, Überdeckung einer Menge durch Mengen kleineren Durchmessers, ''Comment. Math. Helv.'', 18 (1945/46), 73–75; <br/> Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers, 19 (1946/47), 72–73</ref>
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| * For all ''d'' for [[Rotational symmetry|centrally-symmetric]] bodies (A.S. Riesling, 1971).
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| * For all ''d'' for [[Solid of revolution|bodies of revolution]] — the result of Boris Dekster (1995).
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| The problem was finally solved in 1993 by [[Jeff Kahn]] and [[Gil Kalai]], who showed the general answer to the Borsuk's question is ''no''. Their construction shows that {{nobr|''d'' + 1}} pieces do not suffice for {{nobr|1=''d'' = 1,325}} and for each {{nobr|''d'' > 2,014}}.
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| After Andriy V. Bondarenko has shown that Borsuk’s conjecture is false for all {{nobr|''d'' ≥ 65}},<ref>Andriy V. Bondarenko, [http://arxiv.org/abs/1305.2584 On Borsuk's conjecture for two-distance sets]</ref> the current best bound, due to Thomas Jenrich, is 64.<ref>Thomas Jenrich, [http://arxiv.org/abs/1308.0206 A 64-dimensional two-distance counterexample to Borsuk's conjecture]</ref>
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| Apart from finding the minimum number ''d'' of dimensions such that the number of pieces <math>\alpha(d) > d+1</math> mathematicians are interested in finding the general behavior of <math>\alpha(d)</math> function. Kahn and Kalai show that in general (that is for ''d'' big enough), one needs <math>\alpha(d) \ge (1.2)^\sqrt{d}</math> number of pieces. They also quote the upper bound by [[Oded Schramm]], who showed that for every ''ε'', if ''d'' is sufficiently large, <math>\alpha(d) \le \left(\sqrt{3/2} + \varepsilon\right)^d</math>. The correct order of magnitude of ''α''(''d'') is still unknown (see e.g. Alon's article), however it is conjectured that there is a constant {{nobr|''c'' > 1}} such that <math>\alpha(d) > c^d</math> for all {{nobr|''d'' ≥ 1}}.
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| ==See also==
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| *[[Hadwiger conjecture (combinatorial geometry)|Hadwiger's conjecture]] on covering convex bodies with smaller copies of themselves
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [http://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf ''Drei Sätze über die n-dimensionale euklidische Sphäre''] (German 'Three statements of ''n''-dimensional Euclidean sphere') – original Borsuk's article in [[Fundamenta Mathematicae]], made available by [http://matwbn.icm.edu.pl/index.php?jez=en Polish Virtual Library of Science]
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| * Jeff Kahn and [[Gil Kalai]], [http://arxiv.org/abs/math.MG/9307229 A counterexample to Borsuk's conjecture], ''[[Bulletin of the American Mathematical Society]]'' '''29''' (1993), 60–62.
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| * [[Noga Alon]], [http://arxiv.org/abs/math.CO/0212390 Discrete mathematics: methods and challenges], ''Proceedings of the [[International Congress of Mathematicians]], [[Beijing]] 2002'', vol. 1, 119–135.
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| * Aicke Hinrichs and Christian Richter, [http://users.minet.uni-jena.de/~hinrichs/paper/18/borsuk.pdf New sets with large Borsuk numbers], ''Discrete Math.'' '''270''' (2003), 137–147
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| * Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary, ''[[Mathematical Intelligencer]]'' '''26''' (2004), no. 3, 4–12.
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| * [[Oded Schramm]], Illuminating sets of constant width, ''Mathematika'' '''35''' (1988), 180–199.
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| ==Further reading==
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| * Oleg Pikhurko, ''[http://www.math.cmu.edu/~pikhurko/AlgMet.ps Algebraic Methods in Combinatorics]'', course notes.
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| ==External links==
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| * {{MathWorld|urlname=BorsuksConjecture|title=Borsuk's Conjecture}}
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| [[Category:Disproved conjectures]]
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| [[Category:Discrete geometry]]
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The name of the writer is Luther. Meter reading is exactly where my main earnings comes from but quickly I'll be on my own. Delaware is our birth location. What she loves performing is to perform croquet but she hasn't made a dime with it.
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