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{{Short description|Theorem in topology}}
<!-- The French version of this article is a featured article. Large portions have been translated and inserted here in 2009. -->
'''Brouwer's fixed-point theorem''' is a [[fixed-point theorem]] in [[topology]], named after [[Luitzen Egbertus Jan Brouwer|L. E. J. (Bertus) Brouwer]]. It states that for any [[continuous function]] <math>f</math> mapping a nonempty [[compactness|compact]] [[convex set]] to itself, there is a point <math>x_0</math> such that <math>f(x_0)=x_0</math>. The simplest forms of Brouwer's theorem are for continuous functions <math>f</math> from a closed interval <math>I</math> in the real numbers to itself or from a closed [[Disk (mathematics)|disk]] <math>D</math> to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset <math>K
</math> of [[Euclidean space]] to itself.
Among hundreds of [[fixed-point theorem]]s,<ref>E.g. F & V Bayart ''[http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./p/pointfixe.html Théorèmes du point fixe]'' on Bibm@th.net {{webarchive|url=https://web.archive.org/web/20081226200755/http://www.bibmath.net/dico/index.php3?action=affiche&quoi=.%2Fp%2Fpointfixe.html |date=December 26, 2008 }}</ref> Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the [[Jordan curve theorem]], the [[hairy ball theorem]], the [[invariance of dimension]] and the [[Borsuk–Ulam theorem]].<ref>See page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) {{ISBN|2-13-037495-6}}</ref> This gives it a place among the fundamental theorems of topology.<ref>More exactly, according to Encyclopédie Universalis: ''Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales.'' [http://www.universalis.fr/encyclopedie/T705705/BROUWER_L.htm Luizen Brouwer] by G. Sabbagh</ref> The theorem is also used for proving deep results about [[differential equation]]s and is covered in most introductory courses on [[differential geometry]]. It appears in unlikely fields such as [[game theory]]. In economics, Brouwer's fixed-point theorem and its extension, the [[Kakutani fixed-point theorem]], play a central role in the [[Arrow–Debreu model|proof of existence]] of [[general equilibrium]] in market economies as developed in the 1950s by economics Nobel prize winners [[Kenneth Arrow]] and [[Gérard Debreu]].
The theorem was first studied in view of work on differential equations by the French mathematicians around [[Henri Poincaré]] and [[Charles Émile Picard]]. Proving results such as the [[Poincaré–Bendixson theorem]] requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the {{mvar|''n''}}-dimensional closed ball was first proved in 1910 by [[Jacques Hadamard]]<ref name="hadamard-1910">[[Jacques Hadamard]]: ''[https://archive.org/stream/introductionla02tannuoft#page/436/mode/2up Note sur quelques applications de
==Statement==
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A slightly more general version is as follows:<ref>This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see {{cite book|title=General Equilibrium Analysis: Existence and Optimality Properties of Equilibria|first=Monique|last=Florenzano|publisher=Springer|year=2003|isbn=9781402075124|page=7|url=https://books.google.com/books?id=cNBMfxPQlvEC&pg=PA7|access-date=2016-03-08}}</ref>
:;Convex compact set:Every continuous function from a nonempty [[Convex set|convex]] [[Compact space|compact]] subset ''K'' of a Euclidean space to ''K'' itself has a fixed point.<ref>V. & F. Bayart ''[http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./p/pointfixe.html Point fixe, et théorèmes du point fixe ]'' on Bibmath.net. {{webarchive|url=https://web.archive.org/web/20081226200755/http://www.bibmath.net/dico/index.php3?action=affiche&quoi=.%2Fp%2Fpointfixe.html |date=December 26, 2008 }}</ref>
An even more general form is better known under a different name:
:;[[Schauder fixed point theorem]]:Every continuous function from a nonempty convex compact subset ''K'' of a [[Banach space]] to ''K'' itself has a fixed point.<ref>C. Minazzo K. Rider ''[http://math1.unice.fr/~eaubry/Enseignement/M1/memoire.pdf Théorèmes du Point Fixe et Applications aux Equations Différentielles] {{Webarchive|url=https://web.archive.org/web/20180404001651/http://math1.unice.fr/~eaubry/Enseignement/M1/memoire.pdf |date=2018-04-04 }}'' Université de Nice-Sophia Antipolis.</ref>
==Importance of the pre-conditions==
The theorem holds only for functions that are ''endomorphisms'' (functions that have the same set as the ___domain and codomain) and for nonempty sets that are ''compact'' (thus, in particular, bounded and closed) and ''convex'' (or [[Homeomorphism|homeomorphic]] to convex). The following examples show why the pre-conditions are important.
===The function ''f'' as an endomorphism===
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Consider the function
:<math>f(x) = \frac{x+1}{2},</math>
which is a continuous function from the open interval <math>(
===Convexity===
Convexity is not strictly necessary for
The following example shows that
A formal generalization of
===Notes===
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The Brouwer fixed point theorem was one of the early achievements of [[algebraic topology]], and is the basis of more general [[fixed point theorem]]s which are important in [[functional analysis]]. The case ''n'' = 3 first was proved by [[Piers Bohl]] in 1904 (published in ''[[Journal für die reine und angewandte Mathematik]]'').<ref name=Bohl1904>{{cite journal |first=P. |last=Bohl |title= Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage |journal=J. Reine Angew. Math. |volume=127 |issue=3/4 |pages=179–276 |year=1904 }}</ref> It was later proved by [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] in 1909. [[Jacques Hadamard]] proved the general case in 1910,<ref name="hadamard-1910" /> and Brouwer found a different proof in the same year.<ref name="brouwer-1910" /> Since these early proofs were all [[Constructive proof|non-constructive]] [[indirect proof]]s, they ran contrary to Brouwer's [[intuitionist]] ideals. Although the existence of a fixed point is not constructive in the sense of [[Constructivism (mathematics)|constructivism in mathematics]], methods to [[Approximation theory|approximate]] fixed points guaranteed by Brouwer's theorem are now known.<ref name=Karamardian1977>{{cite book|last1=Karamardian|first1=Stephan|title=Fixed points: algorithms and applications|date=1977|publisher=Academic Press|___location=New York|isbn=978-0-12-398050-2}}</ref><ref name=Istratescu1981>{{cite book|last1=Istrăţescu|first1=Vasile|title=Fixed point theory|date=1981|publisher=D. Reidel Publishing Co.|___location=Dordrecht-Boston, Mass.|isbn=978-90-277-1224-0}}</ref>
===
[[File:Théorème-de-Brouwer-(cond-1).jpg|thumb|right|For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.]]
[[File:Théorème-de-Brouwer-(cond-2).jpg|thumb|left|The theorem applies to any disk-shaped area, where it guarantees the existence of a fixed point.]]
Its solution required new methods. As noted by [[Henri Poincaré]], who worked on the [[three-body problem]], there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."<ref name=methodes>[[Henri Poincaré]] ''Les méthodes nouvelles de la mécanique céleste'' T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.</ref>
He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".<ref>Quotation from [[Henri Poincaré]] taken from: P. A. Miquel ''[http://www.arches.ro/revue/no03/no3art03.htm La catégorie de désordre] {{Webarchive|url=https://web.archive.org/web/20160303205947/http://www.arches.ro/revue/no03/no3art03.htm# |date=2016-03-03 }}'', on the website of l'Association roumaine des chercheurs francophones en sciences humaines</ref>
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He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant [[flow (mathematics)|flow]]?<ref>This question was studied in: {{cite journal |first=H. |last=Poincaré |title=Sur les courbes définies par les équations différentielles |journal=[[Journal de Mathématiques Pures et Appliquées]] |volume=2 |issue=4 |pages=167–244 |year=1886 }}</ref> Poincaré discovered that the answer can be found in what we now call the [[topology|topological]] properties in the area containing the trajectory. If this area is [[compact space|compact]], i.e. both [[closed set|closed]] and [[bounded set|bounded]], then the trajectory either becomes stationary, or it approaches a [[limit cycle]].<ref>This follows from the [[Poincaré–Bendixson theorem]].</ref> Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval ''t''. If the area is a circular band, or if it is not closed,<ref>Multiplication by {{sfrac|1|2}} on ]0, 1[<sup>2</sup> has no fixed point.</ref> then this is not necessarily the case.
To understand differential equations better, a new branch of mathematics was born. Poincaré called it ''analysis situs''. The French [[Encyclopædia Universalis]] defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".<ref>"concerne les propriétés invariantes d'une figure
Poincaré's method was analogous to that of [[Charles Émile Picard|Émile Picard]], a contemporary mathematician who generalized the [[Cauchy–Lipschitz theorem]].<ref>See for example: [[Charles Émile Picard|Émile Picard]] ''[http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires] {{Webarchive|url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf# |archive-url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf |archive-date=2011-07-16 |url-status=live |date=2011-07-16 }}'' Journal de Mathématiques p 217 (1893)</ref> Picard's approach is based on a result that would later be formalised by [[Banach fixed-point theorem|another fixed-point theorem]], named after [[Stefan Banach|Banach]]. Instead of the topological properties of the ___domain, this theorem uses the fact that the function in question is a [[contraction mapping|contraction]].
===First proofs===
At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. [[Piers Bohl]], a [[Latvia]]n mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bohl.html Piers Bohl]''</ref> In 1904 he proved the three-dimensional case of our theorem,<ref name="Bohl1904" /> but his publication was not noticed.<ref>{{cite journal |first1=A. D. |last1=Myskis |first2=I. M. |last2=Rabinovic |title=Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем |trans-title=The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P. G. Bohl |language=ru |journal=Успехи математических наук |volume=10 |issue=3 |year=1955 |pages=188–192 |url=http://mi.mathnet.ru/eng/umn/v10/i3/p179 }}</ref>
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[[Image:John f nash 20061102 2.jpg|thumb|220px|left|[[John Forbes Nash|John Nash]] used the theorem in [[game theory]] to prove the existence of an equilibrium strategy profile.]]
The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called [[fixed-point theory]].<ref>V. I. Istratescu ''Fixed Point Theory. An Introduction'' Kluwer Academic Publishers (new edition 2001) {{isbn|1-4020-0301-3}}.</ref>
Brouwer's theorem is probably the most important.<ref>"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (new edition 2001) {{isbn|1-4020-0301-3}}.</ref> It is also among the foundational theorems on the topology of [[topological manifold]]s and is often used to prove other important results such as the [[Jordan curve theorem]].<ref>E.g.: S. Greenwood J. Cao'' [http://www.math.auckland.ac.nz/class750/section5.pdf
Besides the fixed-point theorems for more or less [[contraction mapping|contracting]] functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the [[Borsuk–Ulam theorem]] says that a continuous map from the ''n''-dimensional sphere to '''R'''<sup>n</sup> has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the [[Lefschetz fixed-point theorem]] provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to [[Banach space]]s.<ref>{{cite journal |first=J. |last=Schauder |title=Der Fixpunktsatz in Funktionsräumen |journal=[[Studia Mathematica]] |volume=2 |year=1930 |pages=171–180 |doi= 10.4064/sm-2-1-171-180|doi-access=free }}</ref> This generalization is known as [[Fixed-point theorems in infinite-dimensional spaces|Schauder's fixed-point theorem]], a result generalized further by S. Kakutani to [[Set-valued function|set-valued functions]].<ref>{{cite journal |first=S. |last=Kakutani |title=A generalization of Brouwer's Fixed Point Theorem |journal= Duke Mathematical Journal|volume=8 |year=1941 |issue=3 |pages=457–459 |doi=10.1215/S0012-7094-41-00838-4 }}</ref> One also meets the theorem and its variants outside topology. It can be used to prove the [[Hartman-Grobman theorem]], which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the [[Central Limit Theorem]]. The theorem can also be found in existence proofs for the solutions of certain [[partial differential equation]]s.<ref>These examples are taken from: F. Boyer ''[http://www.cmi.univ-mrs.fr/~fboyer/ter_fboyer2.pdf Théorèmes de point fixe et applications]'' CMI Université Paul Cézanne (2008–2009) [https://www.webcitation.org/5refXIDvI?url=http://www.cmi.univ-mrs.fr/%7Efboyer/ter_fboyer2.pdf Archived copy] at [[WebCite]] (August 1, 2010).</ref>
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:<math>\operatorname{deg}_p(f) = \sum_{x\in f^{-1}(p)} \operatorname{sign}\,\det (df_x).</math>
The degree is, roughly speaking, the number of "sheets" of the preimage ''f'' lying over a small [[open set]] around ''p'', with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of [[winding number]] to higher dimensions.
The degree satisfies the property of ''homotopy invariance'': let <math>f</math> and <math>g</math> be two continuously differentiable functions, and <math>H_t(x)=tf+(1-t)g</math> for <math>0\le t\le 1</math>. Suppose that the point <math>p</math> is a regular value of <math>H_t</math> for all ''t''. Then <math>\deg_p f = \deg_p g</math>.
If there is no fixed point of the boundary of <math>K</math>, then the function
:<math>g(x)=\frac{x-f(x)}{\sup_{
is well-defined, and
<math>H(t,x) = \frac{x-tf(x)}{\sup_{
defines a homotopy from the [[identity function]] to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so <math>g</math> also has degree one at the origin. As a consequence, the preimage <math>g^{-1}(0)</math> is not empty. The elements of <math>g^{-1}(0)</math> are precisely the fixed points of the original function ''f''.
This requires some work to make fully general. The definition of degree must be extended to singular values of ''f'', and then to continuous functions. The more modern advent of [[homology theory]] simplifies the construction of the degree, and so has become a standard proof in the literature.
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The [[hairy ball theorem]] states that on the unit sphere {{mvar|''S''}} in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field {{mvar|'''w'''}} on {{mvar|''S''}}. (The tangency condition means that {{mvar|'''w'''('''x''') ⋅ '''x'''}} = 0 for every unit vector {{mvar|'''x'''}}.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in {{harvtxt|Milnor|1978}}.
In fact, suppose first that {{mvar|'''w'''}} is ''continuously differentiable''. By scaling, it can be assumed that {{mvar|'''w'''}} is a continuously differentiable unit tangent vector on {{mvar|'''S'''}}. It can be extended radially to a small spherical shell {{mvar|''A''}} of {{mvar|''S''}}. For {{mvar|''t''}} sufficiently small, a routine computation shows that the mapping {{mvar|'''f'''<sub>''t''</sub>}}({{mvar|'''x'''}}) = {{mvar|
If {{mvar|'''w'''}} is only a ''continuous'' unit tangent vector on {{mvar|''S''}}, by the [[Weierstrass approximation theorem]], it can be uniformly approximated by a polynomial map {{mvar|'''u'''}} of {{mvar|''A''}} into Euclidean space. The orthogonal projection on to the tangent space is given by {{mvar|'''v'''}}({{mvar|'''x'''}}) = {{mvar|'''u'''}}({{mvar|'''x'''}}) - {{mvar|'''u'''}}({{mvar|'''x'''}}) ⋅ {{mvar|'''x'''}}. Thus {{mvar|'''v'''}} is polynomial and nowhere vanishing on {{mvar|''A''}}; by construction {{mvar|'''v'''}}/||{{mvar|'''v'''}}|| is a smooth unit tangent vector field on {{mvar|''S''}}, a contradiction.
The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that {{mvar|''n''}} is
:<math>{\mathbf w}({\mathbf x}) = (1 - {\mathbf x}\cdot {\mathbf f}({\mathbf x}))\, {\mathbf x} - (1 - {\mathbf x}\cdot {\mathbf x})\, {\mathbf f}({\mathbf x}).</math>
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:<math>{\mathbf X}({\mathbf x},t)=(-t\,{\mathbf w}({\mathbf x}), {\mathbf x}\cdot {\mathbf w}({\mathbf x})).</math>
By construction {{mvar|'''X'''}} is a continuous vector field on the unit sphere of {{mvar|''W''}}, satisfying the tangency condition {{mvar|'''y'''}} ⋅ {{mvar|'''X'''}}({{mvar|'''y'''}}) = 0. Moreover, {{mvar|'''X'''}}({{mvar|'''y'''}}) is nowhere vanishing (because, if {{var|'''x'''}} has norm 1, then {{mvar|'''x'''}} ⋅ {{mvar|'''w'''}}({{mvar|''x''}}) is non-zero; while if {{mvar|'''x'''}} has norm strictly less than 1, then {{mvar|''t''}} and {{mvar|'''w'''}}({{mvar|'''x'''}}) are both non-zero). This contradiction proves the fixed point theorem when {{mvar|''n''}} is
The advantage of this proof is that it uses only elementary techniques; more general results like the [[Borsuk-Ulam theorem]] require tools from [[algebraic topology]].<ref name="Milnor78">{{harvnb|Milnor|1978}}</ref>
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[[Image:Brouwer fixed point theorem retraction.svg|thumb|right|Illustration of the retraction ''F'']]
Suppose, for contradiction, that a continuous function {{nowrap|''f'' : ''D''<sup>''n''</sup> → ''D''<sup>''n''</sup>}} has ''no'' fixed point. This means that, for every point x in ''D''<sup>''n''</sup>, the points ''x'' and ''f''(''x'') are distinct. Because they are distinct, for every point x in ''D''<sup>''n''</sup>, we can construct a unique ray from ''f''(''x'') to ''x'' and follow the ray until it intersects the boundary ''S''<sup>''n''−1</sup> (see illustration). By calling this intersection point ''F''(''x''), we define a function ''F'' : ''D''<sup>''n''</sup> → ''S''<sup>''n''−1</sup> sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever ''x'' itself is on the boundary, then the intersection point ''F''(''x'') must be ''x''.
Consequently, ''F'' is a special type of continuous function known as a [[retraction (topology)|retraction]]: every point of the [[codomain]] (in this case ''S''<sup>''n''−1</sup>) is a fixed point of ''F''.
Intuitively it seems unlikely that there could be a retraction of ''D''<sup>''n''</sup> onto ''S''<sup>''n''−1</sup>, and in the case ''n'' = 1, the impossibility is more basic, because ''S''<sup>0</sup> (i.e., the endpoints of the closed interval ''D''<sup>1</sup>) is not even connected. The case ''n'' = 2 is less obvious, but can be proven by using basic arguments involving the [[fundamental group]]s of the respective spaces: the retraction would induce a surjective [[group homomorphism]] from the fundamental group of ''D''<sup>2</sup> to that of ''S''<sup>1</sup>, but the latter group is isomorphic to '''Z''' while the first group is trivial, so this is impossible. The case ''n'' = 2 can also be proven by contradiction based on a theorem about non-vanishing [[vector field]]s.
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For ''n'' > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of [[Homology (mathematics)|homology groups]]: the homology ''H''<sub>''n''−1</sub>(''D''<sup>''n''</sup>) is trivial, while ''H''<sub>''n''−1</sub>(''S''<sup>''n''−1</sup>) is infinite [[cyclic group|cyclic]]. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.
The impossibility of a retraction can also be shown using the [[de Rham cohomology]] of open subsets of Euclidean space ''E''<sup>''n''</sup>. For ''n'' ≥ 2, the de Rham cohomology of ''U'' = ''E''<sup>''n''</sup> – (0) is one-dimensional in degree 0 and ''n''
===A proof using Stokes' theorem===
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===A proof by Hirsch===
There is also a quick proof, by [[Morris Hirsch]], based on the impossibility of a differentiable retraction.
R. Bruce Kellogg, Tien-Yien Li, and [[James A. Yorke]] turned Hirsch's proof into a [[Computability|computable]] proof by observing that the retract is in fact defined everywhere except at the fixed points.{{sfn|Kellogg|Li|Yorke|1976}} For almost any point
===A proof using oriented area===
A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If <math>r\colon B\to \partial B</math> is a smooth retraction, one considers the smooth deformation <math>g^t(x):=t r(x)+(1-t)x,</math> and the smooth function
:<math>\varphi(t):=\int_B \det D g^t(x) \, dx.</math>
Differentiating under the sign of integral it is not difficult to check that ''{{prime|φ}}''(''t'') = 0 for all ''t'', so ''φ'' is a constant function, which is a contradiction because ''φ''(0) is the ''n''-dimensional volume of the ball, while ''φ''(1) is zero. The geometric idea is that ''φ''(''t'') is the oriented area of ''g''<sup>''t''</sup>(''B'') (that is, the Lebesgue measure of the image of the ball via ''g''<sup>''t''</sup>, taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter ''t'' passes
===A proof using the game Hex===
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===A proof in a weak logical system===
In [[reverse mathematics]], Brouwer's theorem can be proved in the system [[Weak
==Generalizations==
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==References==
*{{cite journal|mr=0283792|last=Boothby| first=William M.|title=On two classical theorems of algebraic topology|journal=[[Amer. Math. Monthly]]|volume= 78|year=1971|issue=3 |pages=237–249|doi=10.2307/2317520 |jstor=2317520}}
*{{cite book|mr=0861409|last=Boothby|first= William M. |title=
An introduction to differentiable manifolds and Riemannian geometry|edition=Second|series= Pure and Applied Mathematics|volume= 120|publisher= Academic Press|year= 1986|isbn= 0-12-116052-1}}
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==External links==
* [http://www.cut-the-knot.org/do_you_know/poincare.shtml#brouwertheorem Brouwer's Fixed Point Theorem for Triangles] at [[cut-the-knot]]
* {{PlanetMath|BrouwerFixedPointTheorem}}
* [http://www.mathpages.com/home/kmath262/kmath262.htm Reconstructing Brouwer] at MathPages
* [https://web.archive.org/web/20110314100800/http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem Brouwer Fixed Point Theorem] at Math Images.
{{Authority control}}
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