= Brouwer fixed-point theorem
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{title2=$F(x_*)=x_*$}
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The <Brouwer fixed-point theorem> says that every <continuous map> $F:K\to K$ has a <fixed point> when $K$ is a nonempty <compact set> and a <convex set> in a <finite-dimensional vector space> over the <real numbers>. Equivalently, take $K\subseteq\mathbb R^n$ for a finite $n$, with the usual <Euclidean space> structure: there is some $x_*\in K$ such that $F(x_*)=x_*$. The conclusion guarantees at least one <fixed point>, without asserting uniqueness or telling us how quickly to find one. https://courses.csail.mit.edu/6.5440/fall23/lectures/L23_images.pdf[MIT lecture slides on Brouwer's theorem]{ref}
The standard pictures are a <closed ball>, including a <closed unit ball>, and a <simplex>. A filled <disk> qualifies; its bounding <circle> alone does not. A <probability simplex> also qualifies even though its <dimension> is smaller than that of its ambient <Euclidean space>. The hypotheses have concrete meanings: being a <convex set> keeps every <line segment> between allowed points inside $K$, and <compactness> prevents escape or loss of a limiting point. In <Euclidean space>, the <Heine-Borel theorem> identifies <compact sets> with <closed sets> that are <bounded sets>. Crucially, every output must belong to the same $K$ as the input.
For a fully computable example, let $D=\{(u,v):u^2+v^2\leq1\}$ be the filled unit <disk>. Apply a counterclockwise quarter-turn <rotation>, shrink by $0.4$, and add a <translation>:
$$
R_{90}(u,v)=(-v,u),\qquad F(u,v)=0.4R_{90}(u,v)+(0.3,0.1).
$$
This <affine map> is a <continuous map>. Because the <rotation> preserves the <Euclidean norm>, the <triangle inequality> gives, for every $x\in D$,
$$
\|F(x)\|\leq0.4\|x\|+\|(0.3,0.1)\|\leq0.4+\sqrt{0.1}\approx0.716228<1.
$$
Thus the whole image lies strictly inside $D$. More precisely, it is a <disk> of <radius> $0.4$ centred at $(0.3,0.1)$. This containment check is what makes the <Brouwer fixed-point theorem> applicable; a formula being a <continuous map> by itself would not suffice.
The <fixed point> can now be found by a <system of linear equations>:
$$
u=-\frac25v+\frac3{10},\qquad v=\frac25u+\frac1{10},\qquad x_*=\left(\frac{13}{58},\frac{11}{58}\right)\approx(0.224138,0.189655).
$$
Indeed, substitution gives $F(x_*)=(-22/290+87/290,26/290+29/290)=x_*$. Its <Euclidean norm> is $\sqrt{290}/58\approx0.293610$, safely inside $D$. The figure shows arrows from sample inputs to their outputs; an arrow disappears at the <fixed point> because the displacement $F(x)-x$ vanishes there.
\Image[/brouwer-disk-map.png]
{title=A quarter-turn, shrinking, and translation keep a disk inside itself}
Repeated <iterations of a map> happen to succeed for this example. Starting at $x_0=(1,0)$ and setting $x_{k+1}=F(x_k)$ gives $x_1=(0.3,0.5)$, $x_2=(0.1,0.22)$, and $x_3=(0.212,0.14)$. Subtracting the <fixed point> equation yields $x_{k+1}-x_*=0.4R_{90}(x_k-x_*)$, so the <Euclidean distance> to $x_*$ shrinks by exactly $0.4$ each step. This extra fact follows from the <contraction mapping> property and the <Banach fixed-point theorem>. It is additional information beyond the <Brouwer fixed-point theorem>.
The distinction becomes visible with two simpler <continuous maps> of the same <disk>. The pure <rotation> $G(u,v)=(-v,u)$ has exactly one <fixed point>, $(0,0)$, since $u=-v$ and $v=u$ force both to vanish. Yet <iterations of a map> starting at $(1,0)$ here cycle through $(0,1)$, $(-1,0)$, and $(0,-1)$ forever. Their <Euclidean distance> from $(0,0)$ stays one, so there is no <convergence in a metric space>. The <reflection> $H(u,v)=(u,-v)$ has an entire <line segment> of <fixed points>, namely $\{(u,0):-1\leq u\leq1\}$. Starting off that <line segment> produces an alternating pair. Existence, uniqueness, and convergence of <iterations of a map> are separate questions.
\Image[/fixed-point-existence-versus-iteration.png]
{title=Existence of a fixed point does not imply uniqueness or convergence of iteration}
Removing hypotheses changes the answer. Restrict the same quarter-turn <rotation> to the <circle> $S^1=\{(u,v):u^2+v^2=1\}$. It remains a <continuous map> into that <compact set>, but its only possible <fixed point>, $(0,0)$, has been removed. The <circle> is not a <convex set>. Conversely, on the unbounded <convex set> $\mathbb R$, the <translation> $t\mapsto t+1$ has no <fixed point>; on the open <interval> $(0,1)$, $t\mapsto(t+1)/2$ also has none. Neither domain is a <compact set>. The finite-dimensional <closed ball> statement should likewise not be extended to arbitrary <infinite-dimensional vector spaces>, where a <closed ball> need not be a <compact set>.
The one-dimensional version explains the connection to the <intermediate value theorem>. For a <continuous map> $f:[a,b]\to[a,b]$, put $h(t)=f(t)-t$. Then $h(a)\geq0$ and $h(b)\leq0$, so the <intermediate value theorem> gives $h(c)=0$ for some $c\in[a,b]$. In higher <dimensions>, the <Brouwer fixed-point theorem> supplies a corresponding existence principle when this scalar sign argument is unavailable. Its practical value is ensuring that a proposed equation $x=F(x)$ has a solution once a suitable domain is found; it supplies no general promise of a fast <algorithm>.
A useful application is <positive matrix eigenvector from simplex normalization>: rescale the output of a strictly positive <matrix> to make its entries sum to one. For example, take
$$
A=\begin{pmatrix}2&1\\1&2\end{pmatrix},\qquad x=(t,1-t),\quad 0\leq t\leq1.
$$
On this <probability simplex>, $Ax=(1+t,2-t)$ has entry sum $3$, so the normalized <continuous map> is $T(x)=((1+t)/3,(2-t)/3)$. Both entries are positive and sum to one, keeping the image inside the <probability simplex>. Its <fixed point> equation is $t=(1+t)/3$, hence $t=1/2$. Consequently $x_*=(1/2,1/2)$ satisfies $Ax_*=(3/2,3/2)=3x_*$: a strictly positive <eigenvector> with <eigenvalue> $3$. In larger <dimensions>, the <Brouwer fixed-point theorem> guarantees the corresponding positive <eigenvector> without requiring us to solve a <system of linear equations> in advance. The <fixed point> expresses preservation of relative weights, rather than preservation of their unnormalized size.
Proofs can be approached through the <no-retraction theorem> or <Sperner's lemma>. In the latter approach, increasingly fine <triangulations> locate approximate <fixed points>; <compactness> supplies a <convergent subsequence>, and <continuous maps> turn its limit into an exact <fixed point>. https://www.mit.edu/~6.7980/brouwer.html[Daskalakis and Farina, Brouwer and Sperner]{ref}
A primary historical reference is L. E. J. Brouwer's 1911 paper Über Abbildung von Mannigfaltigkeiten, whose digitized original is hosted by Zenodo under DOI 10.1007/BF01456812. This gives a direct route to the early literature behind the <Brouwer fixed-point theorem>. https://zenodo.org/records/1428286[Brouwer, Über Abbildung von Mannigfaltigkeiten, 1911]{ref}
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