Talk:Brouwer fixed-point theorem: Difference between revisions

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== Accessible proof==

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<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:198.144.199.xxx|198.144.199.xxx]] ([[User talk:198.144.199.xxx|talk]] • [[Special:Contributions/198.144.199.xxx|contribs]]) 30 August 2001</span></small><!-- Template:Unsigned -->

== FirstCitation proved by Bol?style ==

According to Lyusternik ''Convex Figures and Polyhedra'', the theorem was first proved by a Lettish mathematician named Bol. No references are provided. Anyone know what this is about?--[[User:192.35.35.36|192.35.35.36]] 00:08, 18 Feb 2005 (UTC)

==Proofs==

Why is wikipedia not the place to reroduce a long proof? <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:130.238.5.5|130.238.5.5]] ([[User talk:130.238.5.5|talk]] • [[Special:Contributions/130.238.5.5|contribs]]) 22 September 2005</span></small><!-- Template:Unsigned -->

:What? [[User:El C|El_C]] 20:10, 22 September 2005 (UTC)

:: See [[Wikipedia:WikiProject Mathematics/Proofs]] and the [[WP:MOSMATH|math style manual]]. [[User:Oleg Alexandrov|Oleg Alexandrov]] 22:44, 22 September 2005 (UTC)

== Hex in multiple dimensions? ==

The article says:

:A quite different proof can be given based on the game of [[Hex_(board_game)|Hex]]. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed point theorem for dimension 2. By considering ''n''-dimensional versions of Hex, one can prove that in general that Brouwer's theorem is equivalent to the "no draw" theorem for Hex.

But how does one play an "''n''-dimensional version of Hex"?

:Consider the original 2D hexboard as being made from a lattice, where one connects the lower left corner of a square of a lattice to the upper right corner with an edge. These added diagonals make it so you can have 6 neighbors instead of 4 (on the original lattice). It's easy to see how to cut out an n by n Hex board from this modified lattice. In general, to create an n x n .... x n board, consider a lattice in R^m (where m is the number of dimensions of the board) and then in each m-dimensional cube add a diagonal. Then cut out a board as in 2D.

:Each player has an opposite pair of sides as before, but now some sides belong to neither player. The game is played the same way as before. It doesn't appear to be so interesting to play, but people have devised other higher dimensional versions which are probably more fun and interesting mathematically. In any case, in the version I described, there can never be a draw, and this no-draw result is equivalent to the Brouwer fixed point theorem. --[[User:C S|C S]][[User talk:C S| (Talk)]] 07:03, 15 January 2006 (UTC)

== elemantary proof with stokes' theorem ==

first, the retraction is given by:

[[Image:Theorem of brouwer-F.png|framed|Illustration of the retraction]]

: <math>F(x):=x + \left( \sqrt{1-|x|^2 + \left\langle x,\frac{x-f(x)}{|x-f(x)|} \right\rangle^2 } - \left\langle x, \frac{x-f(x)}{|x-f(x)|} \right\rangle \right) \frac{x-f(x)}{|x-f(x)|} </math>

and it can be proved using the strokes' theorem for differential forms.

for <math>\omega^{n-1}:= F^1\, dF^2\wedge\cdots\wedge dF^n </math>, <math>d\omega^{n-1} = 0</math>, so you get:

<math> 0 = \int_{D^n} \mathrm d\omega^{n-1} = \int_{S^{n-1}} \omega^{n-1} = \int_{S^{n-1}} x_1 dx^2 \wedge \cdots \wedge dx^n = vol (D^n) \neq 0 </math>. first = because the jacobian is 0 by theorem of implicit functions.

See the german page as example. this could be integrated. ~ibotty

:: Looks a bit messy to me.... [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 16:24, 9 February 2006 (UTC)

:It's not that messy! Don't be scared by the symbols, Oleg :-) I think with a little reworking it would make a fine addition. This is a pretty famous proof. Unfortunately, I take a bit of an issue with "elementary" describing this proof. The proof, as given, only proves Brouwer's theorem for sufficiently smooth maps f, since one needs to take the exterior derivative. Luckily, we can ''homotope'' f to be smooth while still keeping the map fixed point free (we can pick a straight-line homotopy that moves every point less than epsilon, where epsilon is smaller than the minimal distance between x and f(x)); however, this is not so trivial to show and can be regarded as a technicality that makes the entire proof not as elementary. And of course, we could debate, if we wished, whether this whole business of Stoke's theorem and smoothing maps is really more elementary than some simple homology (or homotopy) theory. Personally, I think the only kind of proof of Brouwer that really qualifies as elementary are the ones involving some form of coloring trick, e.g. Sperner's lemma or Hex. --[[User:C S|C S]][[User talk:C S| (Talk)]] 01:46, 2 April 2006 (UTC)

:Hmmm...actually I see that since I last closely perused the article, somebody has added another proof of Brouwer for smooth maps. Per my comment right above, this is actually a proof for continuous maps also; I'll add that to the article. --[[User:C S|C S]][[User talk:C S| (Talk)]] 10:42, 2 April 2006 (UTC)

: Stokes theorem may well be derived from the Brouwer Fixed Point Theorem, so I would be weary of using it. But I am not certian of this, as I have not read the proof. --[[User:Dark Side of the Moon|Dark Side of the Moon]] 17:02, 28 August 2006 (UTC)

:: It's not. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/66.191.126.159|66.191.126.159]] ([[User talk:66.191.126.159|talk]]) 06:20, 21 November 2007 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Outline of Proof ==

I believe that there's an error outline of the proof given - in particular the induced transformation from the disk to its boundary need not be a retraction (the points on the boundary need not be fixed by the transformation). At the same time this is only a small error in that it's true that there's no continuous mapping from the disk to its boundary (retraction or not).

:I'm not sure what you mean. The map described is the one that sends x to the point on the boundary given by following the directed line going from f(x) to x. So if x is on the boundary, it gets fixed. --[[User:C S|C S]][[User talk:C S| (Talk)]] 03:36, 1 May 2006 (UTC)

Sorry, you're completely right, I thought the ray was in the other direction.

:However there is still a problem, why does <math>F(x)</math> first have to be continuous (the above section shows that it is, but this should be in the outline). The biggest problem I see is that <math>F : D^2 \to S^1</math> is not '''necessarily''' surjective! --[[User:Dark Side of the Moon|Dark Side of the Moon]] 16:59, 28 August 2006 (UTC)

::''F'' is necessarily surjective, as it maps every element of <math>S^1</math> to itself. As for continuity, this follows from the explicit formula for ''F''(''x'') that someone has given above. --[[User:Zundark|Zundark]] 17:57, 28 August 2006 (UTC)

:::I was mistaken; I had misread and thought that the point ''F(x)'' was created by moving from ''x'' towards ''f(x)'' which would cause problems. The other way, the way it is in the article, will infact work. --[[User:Dark Side of the Moon|Dark Side of the Moon]] 21:10, 28 August 2006 (UTC)

== Non-constructive ==

The history section seems to imply that the theorem does not have a constructive proof. Is there a way to formalize (and prove) this statement? [[User:AxelBoldt|AxelBoldt]] 05:37, 9 June 2006 (UTC)

:Ok, I found in [http://eom.springer.de/B/b017670.htm] that there are in fact algorithms to approximate a fixed point. [[User:AxelBoldt|AxelBoldt]] 03:34, 12 June 2006 (UTC)

== Minor point ==

It seems to me that in the "Notes" near the beginning, where it mentions that the fixed point theorem also applies to objects homeomorphic to the closed unit n-disc, the property "bounded" should be removed from the parentheses. Of course that would be true if the homeomorphism is into some R^m, by the Heine-Borel Theorem. But the homeomorphism need not be to some R^m; it needn't even be to a metric space. Maybe in parentheses, instead of saying "closed, bounded," it should just say "compact." [[User:Kier07|Kier07]] 22:56, 12 December 2006 (UTC)

:The homeomorphic space might not be metric, but it is compact metrizable, and in a compact metrizable space, every metric is bounded. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/66.191.126.159|66.191.126.159]] ([[User talk:66.191.126.159|talk]]) 06:36, 21 November 2007 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Induced map goes the wrong way ==

Since the retraction maps D² to S¹, its induced map goes from &pi;₁(D²) to &pi;₁(S¹) rather than the other way around. The injective map from &pi;₁(S¹) to &pi;₁(D²) is induced by the inclusion, while the retraction induces a surjective map in the other direction. [[User:69.234.55.7|69.234.55.7]] 20:45, 15 November 2007 (UTC)

== Hirsch's proof ==

The claim that leads to the contradiction is not true as stated: it can happen that the inverse image of a regular value of ''F'' is a topological circle in ''D''^''n'' with one point on the boundary sphere ''S''^{''n''&minus;1}. This is compatible with the map ''F'' being a retraction. What then? [[User:Arcfrk|Arcfrk]] ([[User talk:Arcfrk|talk]]) 09:49, 23 March 2008 (UTC)

:um, are you trying to cause trouble? :-) Anyway, I fixed it up a bit to avoid your objection. --[[User:C S|C S]] ([[User talk:C S|talk]]) 18:02, 27 July 2008 (UTC)

==General editing==

Some of the HTML was weird, and I went through and fixed these while reducing some of the spacing in superscripts. A few sentences were written in a way which could have been a little more encyclopaedic, and I removed some plural third-person references where I was able to. There could still be a little bit of a tidy-up on this article, and I certainly might not have done all of the edits correctly since there were a number of them. [[User:Xantharius|Xantharius]] ([[User talk:Xantharius|talk]]) 17:29, 17 April 2008 (UTC)

== Another proof in the case n=2 ==

"The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields."

Why not write the name of the theorem, or at least give a link to it? --[[User:SuneJ|SuneJ]] ([[User talk:SuneJ|talk]]) 15:32, 5 August 2008 (UTC)

== The Liquid in a glass example ==

should either be clarified or deleted. It certainly isn't true as stated. I suppose that whoever wrote it was thinking of continuum fluid dynamics but real fluids aren't continua, they're made of atoms all of which undergo thermal motion. <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/65.19.15.202|65.19.15.202]] ([[User talk:65.19.15.202|talk]]) 15:43, 24 April 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Well that is about how literal you want to be in describing "the liquid" in general as a "way of understanding" it is correct - a point need not be an atom in a mathematical view of the liquid. Although maybe not the exact physical interpretation. Honestly i really don't see the problem here. [[User:Gillis|Gillis]] ([[User talk:Gillis|talk]]) 21:57, 24 April 2009 (UTC)

A point need not be an atom in a mathematical view of the liquid? Although maybe not the exact physical interpretation? This is what I would call a '''clarification''', which is exactly what the OP requested. That's the problem here. It's a very small problem, certainly. And most readers can probably understand the intended concept the way it is stated. Perhaps the word cocktail could be prefaced with (mathematically idealized) as a way to head off any argument about discrete molecules. I really believe such a minor mod would reduce the frustration some readers feel when we encounter apparent disconnects from reality such as this. Thanks for your time.--[[User:Twixter|Twixter]] ([[User talk:Twixter|talk]]) 00:26, 13 July 2012 (UTC)

== One of the statements in the "Intuitive proof" contains an error ==

I have removed the statement from the intuitive proof that claims that the indicated zero set has to contain a line. It does not -- it is an easy exercise, for instance, to construct functions of the sort indicated whose zero sets are things like the "topologist's sine curve", which definitely does not contain a line!

Making the "intuitive proof" precise is actually quite difficult. One would need to know something about dimension theory, but the usual proofs of the basic facts in dimension theory needed to prove the needed results actually use Brouwer's fixed point theorem! Someone should write a warning in the beginning of it to this effect, but I don't have enough free time to do so. <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/24.41.92.61|24.41.92.61]] ([[User talk:24.41.92.61|talk]]) 04:27, 6 July 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

This article mixes parenthetical referencing with footnoted references. The parenthetical ones were there first, so according to [[WP:CITEVAR]] we'd have to use that until explicit consensus. However, it would be significantly easier to turn the couple of parenthetical ones into footnotes than about 50 footnotes into parentheticals. Can we form consensus to continued using footnoted references? <span style="font-family: serif; letter-spacing: 0.1em">–&nbsp;[[User:Finnusertop|Finnusertop]]</span> ([[User talk:Finnusertop|talk]] ⋅ [[Special:Contributions/Finnusertop|contribs]]) 19:59, 24 February 2019 (UTC)

==Illustration==

Since we require a resolution that is high of the graphpaper why bother saying its a graph paper, just makes things confusing?

:I'm sure that'd be okay here. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 00:04, 25 February 2019 (UTC)

Assume that you have four squares on a paper i.e. one centerpoint, now it is easy to place the paper so that it is not directly above a point? <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/85.24.185.96|85.24.185.96]] ([[User talk:85.24.185.96|talk]]) 18:04, 4 September 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

::Great. I've turned the remaining parentheticals into footnotes. <span style="font-family: serif; letter-spacing: 0.1em">–&nbsp;[[User:Finnusertop|Finnusertop]]</span> ([[User talk:Finnusertop|talk]] ⋅ [[Special:Contributions/Finnusertop|contribs]]) 00:10, 25 February 2019 (UTC)

== Function mapping in closedness section ==

== Image problem ==

It is stated that the function f(x) = (x+1)/2 is a continous function from the open interval (-1,1) to itself.

[[:File:Brouwer_fixed_point_theorem_retraction.svg]] isn't displaying properly for me. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 00:05, 19 November 2009 (UTC)

:For meIs it's finenot boththe withcase Firefoxthat 3.5.5the andfunction withmaps Internetfrom Explorer 8. Such problems can be highly browser dependent and are very hard(-1,1) to debug.(0,1)? [[User:Hans AdlerSalomonaber|HansSalomonaber]] ([[User talk:Hans AdlerSalomonaber|Adlertalk]]) 00:1113, 1911 NovemberMarch 20092020 (UTC)

::I'm having problems in Konqueror 3.5.5 and Iceweasel 2.0.0.8. The svg file also doesn't render properly in inkscape. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 02:44, 19 November 2009 (UTC)

::Also doesn't work in Safari 4.0.3 and Firefox 3.0.15 (on a MAC running OSX 10.3). [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 02:47, 19 November 2009 (UTC)

:::Maybe its a locale issue? The image was uploaded on [[de:]] first. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 02:59, 19 November 2009 (UTC)

:::Same problem in Google Chrome. (FYI: the problem I am seeing is that the parentheses appear before ''F'' and ''f'', rather than around the function argument.) [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 18:37, 19 November 2009 (UTC)

::::Ah, since you didn't say what it was I thought it was something more obvious. I had the same problem with my work computer, which was getting a generated PNG file rather than the original SVG. The SVG rendered fine on my platform. I think I have fixed it. [[User:Hans Adler|Hans]] [[User talk:Hans Adler|Adler]] 19:01, 19 November 2009 (UTC)

:::::Fabulous. That did the trick. :-) [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 02:57, 20 November 2009 (UTC)

:It doesn't claim (nor is it required) that the function is [[surjective]], so what's there is correct and appropriate. The example could have even arranged for a bijection, but I don't think it matters much either way. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 00:32, 11 March 2020 (UTC)

== non-constructive, again ==

== Highly skeptical that the remarks "said to have [been] added" by Brouwer are actually due to him ==

"While Brouwer preferred constructive proofs, ironically, the original proofs of his great topological theorems were not constructive, and it took until 1967 for constructive proofs to be found."

<blockquote>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."</blockquote>

Well, in fact there is no constructive proof of the fixed-point theorem: it is false (not realisable)! And, as I understand Brouwer eventually knew a refutation of it, and renounced his own theorem. What we have proofs of is constructive substitutes: there is a fixed point up to epsilon, but this does not always give an exact fixed point because the approximations do not always converge as epsilon goes to zero. --[[Special:Contributions/99.245.206.188|99.245.206.188]] ([[User talk:99.245.206.188|talk]]) 03:32, 21 January 2010 (UTC) (PS: why does an article on Brouwer's fixed point theorem start with a picture of Poincaré?)

The citation is apparently from a French-language educational TV show (https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html). The remarks appear to be spoken by a fictional Brouwer trying to explain his result. The web page that this refers to gives no citation.

== Generalization for compact connected orders ==

I would like to know who originally came up with the "crumpled paper theorem" explanation of the BFPT. It could have been Brouwer himself, but my guess is it was not. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Natkuhn|Natkuhn]] ([[User talk:Natkuhn#top|talk]] • [[Special:Contributions/Natkuhn|contribs]]) 01:06, 8 May 2020 (UTC)</small> <!--Autosigned by SineBot-->

Under "Generalizations" it is stated that if <math>L</math> is a compact, connected order topology, then any continuous function from <math>L^n</math> to itself has a fixed point, but there is no reference given. Does anyone have a proof for this? -- [[User:AndreasF82|AndreasF82]] ([[User talk:AndreasF82|talk]]) 12:57, 20 October 2010 (UTC)

:Oops, yeah, that's a good catch. This probably deserves some looking into. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 01:19, 8 May 2020 (UTC)

== Notes==

Correct me if i'm wrong, but I think that there is an error in the counterexample for domains which aren't closed.

You claim that for the unit disc, which is not closed: <math>f(x,y) = \textstyle (\frac{1}{2}(x+\sqrt{1-y^2}),y)</math>

is a function with no fixed point. But if we set <math>x = \frac{1}{2}</math> and <math>y = \sqrt{\frac{3}{4}}</math>,

then <math> f(x,y) = (x,y) </math>.--[[Special:Contributions/84.73.188.172|84.73.188.172]] ([[User talk:84.73.188.172|talk]]) 18:07, 19 January 2011 (UTC)

== Did Brouwer offer the first proof for continuous functions? ==

:Congratulations! You found an error that has been in the article since [http://en.wikipedia.org/w/index.php?title=Brouwer_fixed_point_theorem&action=historysubmit&diff=57654312&oldid=46581414 June 2006]!

:The idea is valid, though. I will replace the example with a hopefully correct one using the homeomorphism of the interval (-π/2,π/2) with the real line via the tan and arctan functions. Since I have just made this up and did not put much thought into it, it would be nice if you could check it. [[User:Hans Adler|Hans]] [[User talk:Hans Adler|Adler]] 20:46, 19 January 2011 (UTC)

In the book ''History of Topology'' by James on pages 273-274: "Bohl's theorem is also equivalent to the Brouwer theorem. Bohl's theorem was published in 1904, with a proof that required that f be differentiable. Brouwer published his fixed point theorem, for continuous functions on the 3-ball, in 1909. When the first proof for the n-ball, with f '''differentiable''', appeared in print a year later, in an appendix by '''J. Hadamard''' to a text by Tannery, the theorem was called the 'Brouwer Fixed Point Theorem', which suggests that the result was already famous by that time. It is not known in what year Brouwer made his discovery and, apparently, communicated it to other mathematicians in an informal manner. The first published proof of the general case, that is, for '''continuous''' functions on the n-ball, was by '''Brouwer''' himself in 1912."

::Yes, I guess the corrected version is ok. Another, maybe simpler example, would be the open interval (0,1) and the function <math> f(x) = \frac{x}{2} </math>. Since

::the function doesn't have to be bijective, this should also work out. --[[Special:Contributions/84.73.188.172|84.73.188.172]] ([[User talk:84.73.188.172|talk]]) 14:28, 22 January 2011 (UTC)

In the book ''Brouwer Degree. The Core of Nonlinear Analysis'' by Mawhin on page 393: "In 1910, Jules Tannery published the second volume of the second edition of his book ‘Introduction à la théorie des fonctions de variables réelles’, for the time and still now a very modern presentation of analysis, introducing Weierstrass’ rigor in France. This volume two ended with a Note of Jacques Hadamard, connected in the following way to Tannery’s book material: 'The proof, following M. Ames, of Jordan’s theorem on closed curves without double point is based upon the concept of order of a point or, equivalently on the consideration of the variation of the argument. The generalization to the case where the dimension is larger than two is given by the Kronecker index. It is a now classical notion, mainly since the publication of the Traité d’Analyse of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications in various recent works. My aim is to present here some of them. All the following reasonings [...] only use the continuity of the considered functions.'"

== Illustrations: Ordinary map of a country ==

The previous quote from Mawhin's book has a strange omission ([...]) that may indicate that Hadamard did not write down a proof for the continuous case. '''Can someone please check out if Hadamard proved the general case for differentiable functions and not for continuous functions?'''

"Similarly: 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." A country is not necessarily convex (can you name one that is?), which is required in the Brouwer's fixed-point theorem, so I don't see how this is an illustration of the theorem. An ordinary map is usually a [[contraction mapping]], so this can be proved using the [[Banach fixed-point theorem]] instead. --[[Special:Contributions/82.130.37.20|82.130.37.20]] ([[User talk:82.130.37.20|talk]]) 18:07, 7 February 2012 (UTC)

:On the other hand, if the table is convex (or more generally, if the map is inside a convex subset of the country), the Brouwer's fixed-point theorem can be used to prove that there is a fixed point in the part of the map that represents the table (or the convex subset). But this is not as simple as the example currently given in the article. --[[Special:Contributions/82.130.37.20|82.130.37.20]] ([[User talk:82.130.37.20|talk]]) 18:24, 7 February 2012 (UTC)

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:'''Comments.''' Please sign your posts with <nowiki>~~~~</nowiki> and, per [[WP:TPG]], add comments at the ''end'' of the talk page. Please stop edit-warring to insert your own [[WP:POV|point of view]]. There is already a discussion of the history of the FPT in the section [[Brouwer fixed-point theorem#First proofs]]. Bohl's proof applied to three dimensions. There are several historical accounts of the FPT, notably "A history of algebraic and differential topology, 1900–1960" by [[Jean Dieudonné]]. [[User:Mathsci|Mathsci]] ([[User talk:Mathsci|talk]]) 17:43, 21 February 2022 (UTC)

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