Lusternik–Schnirelmann theorem
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In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows.
If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points.
It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.[1][2]
Equivalent results
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result can be reduced to the other result in its column.[3]
| Algebraic topology | Combinatorics | Set covering |
|---|---|---|
| Brouwer fixed-point theorem | Sperner's lemma | KKM lemma |
| Borsuk–Ulam theorem | Tucker's lemma | Lusternik–Schnirelmann theorem |
References
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- ↑ Lua error in package.lua at line 80: module 'strict' not found.. Bollobás (2006) cites pp. 26–31 of this 68-page pamphlet for the theorem.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
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