en>AvicAWB: clean up using AWB

2011-03-01T03:25:47Z

clean up using AWB

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[[File:TuckerLemExample.png|thumb|350px|In the this example, where n=2, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist.]]

In [[mathematics]], '''Tucker's lemma''' is a [[combinatorics|combinatorial]] analog of the [[Borsuk–Ulam theorem]], named after [[Albert W. Tucker]].

Let T be a [[Triangulation_(topology)|triangulation]] of the closed n-dimensional [[Ball_(mathematics)|ball]] ''B''''n''. Assume T is antipodally symmetric on the boundary [[sphere]] '''S'''''n-1''. That means that the subset of [[Simplex|simplices]] of T which are in '''S'''''n-1'' provides a triangulation of '''S'''''n-1'' where if σ is a simplex then so is −σ. Let
:<math>L:V(T)\to\{+1,-1,+2,-2,...,+n,-n\}</math>
be a labelling of the vertices of T which satisfies L(−v)=−L(v) for all vertices v in '''S'''''n-1''. Then Tucker's lemma states that there exists a 1-simplex in T whose vertices are labelled by the same number but with opposite signs.

== See also ==
* [[Brouwer fixed point theorem]]
* [[Sperner's lemma]]
* [[Topological combinatorics]]

==References==
*{{citation
| last1 = Freund | first1 = Robert M.
| last2 = Todd | first2 = Michael J.
| doi = 10.1016/0097-3165(81)90027-3
| issue = 3
| journal = [[Journal of Combinatorial Theory]]
| mr = 618536
| pages = 321–325
| series = Series A
| title = A constructive proof of Tucker's combinatorial lemma
| volume = 30
| year = 1981}}.
*{{citation
| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
| isbn = 3-540-00362-2
| page = 34
| publisher = [[Springer-Verlag]]
| title = Using the Borsuk–Ulam Theorem
| year = 2003}}.
*{{citation
| last = Tucker | first = A. W. | author-link = Albert W. Tucker
| contribution = Some topological properties of disk and sphere
| location = Toronto
| mr = 0020254
| pages = 285–309
| publisher = University of Toronto Press
| title = Proc. First Canadian Math. Congress, Montreal, 1945
| year = 1946}}.

[[Category:Combinatorics]]
[[Category:Topology]]
[[Category:Lemmas]]

{{combin-stub}}

Generalized semi-infinite programming - Revision history

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