Brouwer's fixed-point theorem is afixed-point theorem intopology, named afterL. E. J. (Bertus) Brouwer. It states that for anycontinuous function${\displaystyle f}$ mapping a nonemptycompactconvex set to itself, there is a point${\displaystyle x_0}$ such that${\displaystyle f(x_0)=x_0}$. The simplest forms of Brouwer's theorem are for continuous functions${\displaystyle f}$ from a closed interval${\displaystyle I}$ in the real numbers to itself or from a closeddisk${\displaystyle D}$ to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset${\displaystyle K}$ ofEuclidean space to itself.

Among hundreds offixed-point theorems,^[1] 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 theJordan curve theorem, thehairy ball theorem, theinvariance of dimension and theBorsuk–Ulam theorem.^[2] This gives it a place among the fundamental theorems of topology.^[3] The theorem is also used for proving deep results aboutdifferential equations and is covered in most introductory courses ondifferential geometry. It appears in unlikely fields such asgame theory. In economics, Brouwer's fixed-point theorem and its extension, theKakutani fixed-point theorem, play a central role in theproof of existence ofgeneral equilibrium in market economies as developed in the 1950s by economics Nobel prize winnersKenneth Arrow andGérard Debreu.

The theorem was first studied in view of work on differential equations by the French mathematicians aroundHenri Poincaré andCharles Émile Picard. Proving results such as thePoincaré–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 then-dimensional closed ball was first proved in 1910 byJacques Hadamard^[4] and the general case for continuous mappings by Brouwer in 1911.^[5]

The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

In the plane

Everycontinuous function from acloseddisk to itself has at least one fixed point.^[6]

This can be generalized to an arbitrary finite dimension:

In Euclidean space

Every continuous function from aclosed ball of aEuclidean space into itself has a fixed point.^[7]

A slightly more general version is as follows:^[8]

Convex compact set

Every continuous function from a nonemptyconvexcompact subsetK of a Euclidean space toK itself has a fixed point.^[9]

An even more general form is better known under a different name:

Schauder fixed point theorem

Every continuous function from a nonempty convex compact subsetK of aBanach space toK itself has a fixed point.^[10]

The theorem holds only for functions that areendomorphisms (functions that have the same set as the domain and codomain) and for nonempty sets that arecompact (thus, in particular, bounded and closed) andconvex (orhomeomorphic to convex). The following examples show why the pre-conditions are important.

Consider the function

${\displaystyle f(x)=x+1}$

with domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism.

Consider the function

${\displaystyle f(x)=x+1,}$

which is a continuous function from${\displaystyle \mathbb{R}}$ to itself. As it shifts every point to the right, it cannot have a fixed point. The space${\displaystyle \mathbb{R}}$ is convex and closed, but not bounded.

Consider the function

${\displaystyle f(x)=\frac{x+1}{2},}$

which is a continuous function from the open interval${\displaystyle (-1,1)}$ to itself. Since the point${\displaystyle x=1}$ is not part of the interval, there is no point in the domain such that${\displaystyle f(x)=x}$. The set${\displaystyle (-1,1)}$ is convex and bounded, but not closed. On the other hand, the function${\displaystyle f}$ does have a fixed point in theclosed interval${\displaystyle [-1,1]}$, namely${\displaystyle x=1}$. The closed interval${\displaystyle [-1,1]}$ is compact, the open interval${\displaystyle (-1,1)}$ is not.

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant underhomeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball${\displaystyle D^n}$. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore alsoclosed, bounded,connected,without holes, etc.).

The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function${\displaystyle f(x)=-x}$, which is a continuous function from the unit circle to itself. Since-x≠x holds for any point of the unit circle,f has no fixed point. The analogous example works for then-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex) . The functionfdoes have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from theLefschetz fixed-point theorem.^[11]

The continuous function in this theorem is not required to bebijective orsurjective.

The theorem has several "real world" illustrations. Here are some examples.

1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of then = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.
2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
3. In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktailshaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.^{[citation needed]}

The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee.^[12]If one stirs to dissolve a lump of sugar, it appears there is always a point without motion.He drew the conclusion that at any moment, there is a point on the surface that is not moving.^[13]The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit.The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."^[13]Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such asStefan Banach's, that guarantee uniqueness.

In one dimension, the result is intuitive and easy to prove. The continuous functionf is defined on a closed interval [a, b] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [a, b] which mapsx tox (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the functiong which mapsx tof(x) − x. It is ≥ 0 ona and ≤ 0 on b. By theintermediate value theorem,g has azero in [a, b]; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."^[13]

The Brouwer fixed point theorem was one of the early achievements ofalgebraic topology, and is the basis of more generalfixed point theorems which are important infunctional analysis. The casen = 3 first was proved byPiers Bohl in 1904 (published inJournal für die reine und angewandte Mathematik).^[14] It was later proved byL. E. J. Brouwer in 1909.Jacques Hadamard proved the general case in 1910,^[4] and Brouwer found a different proof in the same year.^[5] Since these early proofs were allnon-constructiveindirect proofs, they ran contrary to Brouwer'sintuitionist ideals. Although the existence of a fixed point is not constructive in the sense ofconstructivism in mathematics, methods toapproximate fixed points guaranteed by Brouwer's theorem are now known.^[15][16]

At the end of the 19th century, the old problem^[17] of thestability of the solar system returned into the focus of the mathematical community.^[18]Its solution required new methods. As noted byHenri Poincaré, who worked on thethree-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."^[19]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".^[20]

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 constantflow?^[21] Poincaré discovered that the answer can be found in what we now call thetopological properties in the area containing the trajectory. If this area iscompact, i.e. bothclosed andbounded, then the trajectory either becomes stationary, or it approaches alimit cycle.^[22] 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,^[23] then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called itanalysis situs. The FrenchEncyclopædia 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".^[24] In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,^[25] although the connection with the subject of this article was not yet apparent.^[26] A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as thefundamental group or sometimes thePoincaré group.^[27] This method can be used for a very compact proof of the theorem under discussion.

Poincaré's method was analogous to that ofÉmile Picard, a contemporary mathematician who generalized theCauchy–Lipschitz theorem.^[28] Picard's approach is based on a result that would later be formalised byanother fixed-point theorem, named afterBanach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is acontraction.

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, aLatvian mathematician, applied topological methods to the study of differential equations.^[29] In 1904 he proved the three-dimensional case of our theorem,^[14] but his publication was not noticed.^[30]

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especiallymathematical logic andtopology. His initial interest lay in an attempt to solveHilbert's fifth problem.^[31] In 1909, during a voyage to Paris, he metHenri Poincaré,Jacques Hadamard, andÉmile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved thehairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.^[32] These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.^[33] The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such ashomotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods.Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."^[34]

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,^[5] as well as other key theorems such as the invariance of dimension.^[35] In the context of this work, Brouwer also generalized theJordan curve theorem to arbitrary dimension and established the properties connected with thedegree of a continuous mapping.^[36] This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs becamealgebraic topology.^[37]

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 calledfixed-point theory.^[38]Brouwer's theorem is probably the most important.^[39] It is also among the foundational theorems on the topology oftopological manifolds and is often used to prove other important results such as theJordan curve theorem.^[40]

Besides the fixed-point theorems for more or lesscontracting 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, theBorsuk–Ulam theorem says that a continuous map from then-dimensional sphere toRⁿ has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, theLefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized toBanach spaces.^[41] This generalization is known asSchauder's fixed-point theorem, a result generalized further by S. Kakutani toset-valued functions.^[42] One also meets the theorem and its variants outside topology. It can be used to prove theHartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of theCentral Limit Theorem. The theorem can also be found in existence proofs for the solutions of certainpartial differential equations.^[43]

Other areas are also touched. Ingame theory,John Nash used the theorem to prove that in the game ofHex there is a winning strategy for white.^[44] In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets.^[45]

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems arenot constructive,^[46] and Brouwer's dissatisfaction with this is partly what led him to articulate the idea ofconstructivity. He became the originator and zealous defender of a way of formalising mathematics that is known asintuitionism, which at the time made a stand againstset theory.^[47] Brouwer disavowed his original proof of the fixed-point theorem.

Brouwer's original 1911 proof relied on the notion of thedegree of a continuous mapping, stemming from ideas indifferential topology. Several modern accounts of the proof can be found in the literature, notablyMilnor (1965).^[48][49]

Let${\displaystyle K=\overline{B(0)}}$ denote the closed unit ball in${\displaystyle {\mathbb{R}}^n}$ centered at the origin. Suppose for simplicity that${\displaystyle f:K\to K}$ is continuously differentiable. Aregular value of${\displaystyle f}$ is a point${\displaystyle p\in B(0)}$ such that theJacobian of${\displaystyle f}$ is non-singular at every point of the preimage of${\displaystyle p}$. In particular, by theinverse function theorem, every point of the preimage of${\displaystyle f}$ lies in${\displaystyle B(0)}$ (the interior of${\displaystyle K}$). The degree of${\displaystyle f}$ at a regular value${\displaystyle p\in B(0)}$ is defined as the sum of the signs of theJacobian determinant of${\displaystyle f}$ over the preimages of${\displaystyle p}$ under${\displaystyle f}$:

${\displaystyle {\mathrm{deg}}_p(f)=\sum _{x\in f^{-1}(p)}\mathrm{sign}det(d{f}_x).}$

The degree is, roughly speaking, the number of "sheets" of the preimagef lying over a smallopen set aroundp, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization ofwinding number to higher dimensions.

The degree satisfies the property ofhomotopy invariance: let${\displaystyle f}$ and${\displaystyle g}$ be two continuously differentiable functions, and${\displaystyle H_t(x)=tf+(1-t)g}$ for${\displaystyle 0\le t\le 1}$. Suppose that the point${\displaystyle p}$ is a regular value of${\displaystyle H_t}$ for allt. Then${\displaystyle {\mathrm{deg}}_{p}f={\mathrm{deg}}_{p}g}$.

If there is no fixed point of the boundary of${\displaystyle K}$, then the function

${\displaystyle g(x)=\frac{x-f(x)}{\underset{y\in K}{sup}\left|y-f(y)\right|}}$

is well-defined, and

${\displaystyle H(t,x)=\frac{x-tf(x)}{\underset{y\in K}{sup}\left|y-tf(y)\right|}}$

defines a homotopy from theidentity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so${\displaystyle g}$ also has degree one at the origin. As a consequence, the preimage${\displaystyle g^{-1}(0)}$ is not empty. The elements of${\displaystyle g^{-1}(0)}$ are precisely the fixed points of the original functionf.

This requires some work to make fully general. The definition of degree must be extended to singular values off, and then to continuous functions. The more modern advent ofhomology theory simplifies the construction of the degree, and so has become a standard proof in the literature.

Thehairy ball theorem states that on the unit sphereS in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector fieldw onS. (The tangency condition means thatw(x) ⋅x = 0 for every unit vectorx.) 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 inMilnor (1978).

In fact, suppose first thatw iscontinuously differentiable. By scaling, it can be assumed thatw is a continuously differentiable unit tangent vector onS. It can be extended radially to a small spherical shellA ofS. Fort sufficiently small, a routine computation shows that the mappingf_t(x) =x +tw(x) is acontraction mapping onA and that the volume of its image is a polynomial int. On the other hand, as a contraction mapping,f_t must restrict to a homeomorphism ofS onto (1 +t²)^⁠1/2⁠S andA onto (1 +t²)^⁠1/2⁠A. This gives a contradiction, because, if the dimensionn of the Euclidean space is odd, (1 +t²)^n/2 is not a polynomial.

Ifw is only acontinuous unit tangent vector onS, by theWeierstrass approximation theorem, it can be uniformly approximated by a polynomial mapu ofA into Euclidean space. The orthogonal projection on to the tangent space is given byv(x) =u(x) -u(x) ⋅x. Thusv is polynomial and nowhere vanishing onA; by constructionv/||v|| is a smooth unit tangent vector field onS, a contradiction.

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose thatn is even. If there were a fixed-point-free continuous self-mappingf of the closed unit ballB of then-dimensional Euclidean spaceV, set

${\displaystyle \mathbf{w}(\mathbf{x})=(1-\mathbf{x}\cdot \mathbf{f}(\mathbf{x}))\mathbf{x}-(1-\mathbf{x}\cdot \mathbf{x})\mathbf{f}(\mathbf{x}).}$

Sincef has no fixed points, it follows that, forx in theinterior ofB, the vectorw(x) is non-zero; and forx inS, the scalar product
x ⋅w(x) = 1 –x ⋅f(x) is strictly positive. From the originaln-dimensional space Euclidean spaceV, construct a new auxiliary
(n + 1)-dimensional spaceW =V xR, with coordinatesy = (x,t). Set

${\displaystyle \mathbf{X}(\mathbf{x},t)=(-t\mathbf{w}(\mathbf{x}),\mathbf{x}\cdot \mathbf{w}(\mathbf{x})).}$

By constructionX is a continuous vector field on the unit sphere ofW, satisfying the tangency conditiony ⋅X(y) = 0. Moreover,X(y) is nowhere vanishing (because, ifx has norm 1, thenx ⋅w(x) is non-zero; while ifx has norm strictly less than 1, thent andw(x) are both non-zero). This contradiction proves the fixed point theorem whenn is even. Forn odd, one can apply the fixed point theorem to the closed unit ballB inn + 1 dimensions and the mappingF(x,y) = (f(x),0).The advantage of this proof is that it uses only elementary techniques; more general results like theBorsuk-Ulam theorem require tools fromalgebraic topology.^[50]

The proof uses the observation that theboundary of then-diskDⁿ isSⁿ⁻¹, the (n − 1)-sphere.

Suppose, for contradiction, that a continuous functionf :Dⁿ →Dⁿ hasno fixed point. This means that, for every point x inDⁿ, the pointsx andf(x) are distinct. Because they are distinct, for every point x inDⁿ, we can construct a unique ray fromf(x) tox and follow the ray until it intersects the boundarySⁿ⁻¹ (see illustration). By calling this intersection pointF(x), we define a functionF : Dⁿ → Sⁿ⁻¹ sending each point in the disk to its corresponding intersection point on the boundary. As a special case, wheneverx itself is on the boundary, then the intersection pointF(x) must bex.

Consequently,F is a special type of continuous function known as aretraction: every point of thecodomain (in this caseSⁿ⁻¹) is a fixed point ofF.

Intuitively it seems unlikely that there could be a retraction ofDⁿ ontoSⁿ⁻¹, and in the casen = 1, the impossibility is more basic, becauseS⁰ (i.e., the endpoints of the closed intervalD¹) is not even connected. The casen = 2 is less obvious, but can be proven by using basic arguments involving thefundamental groups of the respective spaces: the retraction would induce a surjectivegroup homomorphism from the fundamental group ofD² to that ofS¹, but the latter group is isomorphic toZ while the first group is trivial, so this is impossible. The casen = 2 can also be proven by contradiction based on a theorem about non-vanishingvector fields.

Forn > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use ofhomology groups: the homologyH_n−1(Dⁿ) is trivial, whileH_n−1(Sⁿ⁻¹) is infinitecyclic. 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 thede Rham cohomology of open subsets of Euclidean spaceEⁿ. Forn ≥ 2, the de Rham cohomology ofU =Eⁿ – (0) is one-dimensional in degree 0 andn – 1, and vanishes otherwise. If a retraction existed, thenU would have to be contractible and its de Rham cohomology in degreen – 1 would have to vanish, a contradiction.^[51]

As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retractionF from the ballB onto its boundary ∂B. In that case it can be assumed thatF is smooth, since it can be approximated using theWeierstrass approximation theorem or byconvolving with non-negative smoothbump functions of sufficiently small support and integral one (i.e.mollifying). Ifω is avolume form on the boundary then byStokes' theorem,

giving a contradiction.^[52][53]

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifoldM onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the formω generates thede Rham cohomology groupH^n-1(∂M) which is isomorphic to the homology groupH_n-1(∂M) byde Rham's theorem.^[54]

The BFPT was proved by Knaster-Kuratowski-Mazurkiewicz usingSperner's lemma. We now give an outline of the proof for the special case in whichf is a function from the standardn-simplex,${\displaystyle {\mathrm{\Delta }}^n,}$ to itself, where

${\displaystyle {\mathrm{\Delta }}^n=\left\{P\in {\mathbb{R}}^{n+1}\mid \sum _{i=0}^n{P}_i=1\text{ and }{P}_i\ge 0\text{ for all }i\right\}.}$

For every point${\displaystyle P\in {\mathrm{\Delta }}^n,}$ also${\displaystyle f(P)\in {\mathrm{\Delta }}^n.}$ Hence the sum of their coordinates is equal:

${\displaystyle \sum _{i=0}^n{P}_i=1=\sum _{i=0}^{n}f(P{)}_i}$

Hence, by the pigeonhole principle, for every${\displaystyle P\in {\mathrm{\Delta }}^n,}$ there must be an index${\displaystyle j\in \{0,\dots ,n\}}$ such that the${\displaystyle j}$th coordinate of${\displaystyle P}$ is greater than or equal to the${\displaystyle j}$th coordinate of its image underf:

${\displaystyle P_j\ge f(P{)}_j.}$

Moreover, if${\displaystyle P}$ lies on ak-dimensional sub-face of${\displaystyle {\mathrm{\Delta }}^n,}$ then by the same argument, the index${\displaystyle j}$ can be selected from among thek + 1 coordinates which are not zero on this sub-face.

We now use this fact to construct a Sperner coloring. For every triangulation of${\displaystyle {\mathrm{\Delta }}^n,}$ the color of every vertex${\displaystyle P}$ is an index${\displaystyle j}$ such that${\displaystyle f(P{)}_j\le P_j.}$

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is ann-dimensional simplex whose vertices are colored with the entire set ofn + 1 available colors.

Becausef is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point${\displaystyle P}$ which satisfies the labeling condition in all coordinates:${\displaystyle f(P{)}_j\le P_j}$ for all${\displaystyle j.}$

Because the sum of the coordinates of${\displaystyle P}$ and${\displaystyle f(P)}$ must be equal, all these inequalities must actually be equalities. But this means that:

${\displaystyle f(P)=P.}$

That is,${\displaystyle P}$ is a fixed point of${\displaystyle f.}$

There is also a quick proof, byMorris Hirsch, based on the impossibility of a differentiable retraction. Letf denote a continuous map from the unit ball Dⁿ in n-dimensional Euclidean space to itself and assume thatf fixes no point. By continuity and the fact that Dⁿ is compact, it follows that for some ε > 0, ∥x -f(x)∥ > ε for all x in Dⁿ. Then the mapf can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using theWeierstrass approximation theorem or byconvolving with smoothbump functions. One then defines a retraction as above by sending each x to the point of ∂Dⁿ where the unique ray from x throughf(x) intersects ∂Dⁿ, and this must now be a differentiable mapping. Such a retraction must have a non-singular value p ∈ ∂Dⁿ, bySard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse imagef ^-1(p) would be a compact 1-manifold with boundary. Such a boundary would have to contain at least two endpoints, and these would have to lie on the boundary of the original ball. This would mean that the inverse image of one point on ∂Dⁿ contains a different point on ∂Dⁿ, contradicting the definition of a retraction Dⁿ → ∂Dⁿ.^[55]

R. Bruce Kellogg, Tien-Yien Li, andJames A. Yorke turned Hirsch's proof into acomputable proof by observing that the retract is in fact defined everywhere except at the fixed points.^[56] For almost any pointq on the boundary — assuming it is not a fixed point — the 1-manifold with boundary mentioned above does exist and the only possibility is that it leads fromq to a fixed point. It is an easy numerical task to follow such a path fromq to the fixed point so the method is essentially computable.^[57] gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If${\displaystyle r:B\to \mathrm{\partial }B}$ is a smooth retraction, one considers the smooth deformation${\displaystyle g^t(x):=tr(x)+(1-t)x,}$ and the smooth function

${\displaystyle \phi (t):={\int }_{B}detD{g}^t(x)dx.}$

Differentiating under the sign of integral it is not difficult to check thatφ′(t) = 0 for allt, soφ is a constant function, which is a contradiction becauseφ(0) is then-dimensional volume of the ball, whileφ(1) is zero. The geometric idea is thatφ(t) is the oriented area ofg^t(B) (that is, the Lebesgue measure of the image of the ball viag^t, 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 parametert passes from 0 to 1 the mapg^t transforms continuously from the identity map of the ball, to the retractionr, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area ofr is necessarily 0, as its image is the boundary of the ball, a set of null measure.^[58]

A quite different proof given byDavid Gale is based on the game ofHex. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By consideringn-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to thedeterminacy theorem for Hex.^[59]

The Lefschetz fixed-point theorem says that if a continuous mapf from a finite simplicial complexB to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number

${\displaystyle {\displaystyle \sum _n(-1{)}^n\mathrm{Tr}(f|H_n(B))}}$

and in particular if the Lefschetz number is nonzero thenf must have a fixed point. IfB is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zerosimplicial homology group is:${\displaystyle H_0(B)}$ andf acts as the identity on this group, sof has a fixed point.^[60][61]

Inreverse mathematics, Brouwer's theorem can be proved in the systemWKL₀, and conversely over the base systemRCA₀ Brouwer's theorem for a square implies theweak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem.

The Brouwer fixed-point theorem forms the starting point of a number of more generalfixed-point theorems.

The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitraryHilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are notcompact. For example, in the Hilbert spaceℓ² of square-summable real (or complex) sequences, consider the mapf : ℓ² → ℓ² which sends a sequence (x_n) from the closed unit ball of ℓ² to the sequence (y_n) defined by

${\displaystyle y_0=\sqrt{1-\Vert x{\Vert }_2^2}\text{ and}{y}_n=x_{n-1}\text{ for }n\ge 1.}$

It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ², but does not have a fixed point.

The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption ofconvexity. Seefixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.

There is also finite-dimensional generalization to a larger class of spaces: If${\displaystyle X}$ is a product of finitely many chainable continua, then every continuous function${\displaystyle f:X\to X}$ has a fixed point,^[62] where a chainable continuum is a (usually but in this case not necessarilymetric)compactHausdorff space of which everyopen cover has a finite open refinement${\displaystyle \{U_1,\dots ,U_m\}}$, such that${\displaystyle U_i\cap U_j\ne \mathrm{\varnothing }}$ if and only if${\displaystyle |i-j|\le 1}$. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.

TheKakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays inRⁿ, but considers upperhemi-continuousset-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

TheLefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms ofsingular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case ofDⁿ.

There are several fixed-point theorems which come in three equivalent variants: analgebraic 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 in the top row can be deduced from the one below it in the same column.^[63]

• Banach fixed-point theorem
• Fixed-point computation
• Infinite compositions of analytic functions
• Nash equilibrium
• Poincaré–Miranda theorem – equivalent to the Brouwer fixed-point theorem
• Topological combinatorics

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• Brouwer's Fixed Point Theorem for Triangles atcut-the-knot
• Brouwer fixed-point theorem atPlanetMath.
• Reconstructing Brouwer at MathPages
• Brouwer Fixed Point Theorem at Math Images.

• Fixed-point theorems
• Eponymous theorems of mathematics
• Theory of continuous functions
• Theorems in topology
• Theorems in convex geometry

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