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One of the elementary way to prove of the Brouwer fixed-point theorem is, making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple computation on the "oriented area" of smooth mappings $g:B\subset \mathbb {R}^n\rightarrow\mathbb {R}^n$ $$\int_B \operatorname{det} D g(x) dx$$

and only involves a differentiation under the sign of integral with respect to the parameter of deformation (I mentioned this proof in this wiki-article) . Due to this fact, I sometimes like to use it in elementary courses as a meaningful application of differential calculus and Lebesgue integration (on the other hand, the geometrical ideas behind remain a bit hidden, but that is an other story).

However, a slight annoyance to me now is, that I can't remember where I read this proof the first time, several years ago. I would be very glad to learn a reference, and (if it is known) the name of the inventor of this nice proof.

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    $\begingroup$ I think its come from DUnford Schwartz Linear OPerators I p. 467 $\endgroup$
    – Buschi Sergio
    Commented Sep 25, 2012 at 12:53

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There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, vol. 74, Amer. Math. Soc., 2006

The proof is constructive and it leads to an algorithm for generating a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to

B. Kuratowski, C. Knaster, C. Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n-dimensionale Simplexe, Funt. Math. 14(1929), 132-137.

Note This does not really answer your question.

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    $\begingroup$ It answers the title of the question. :) $\endgroup$
    – Jim Conant
    Commented Sep 24, 2012 at 13:25
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    $\begingroup$ THe KKM paper is freely available at: matwbn.icm.edu.pl/ksiazki/fm/fm14/fm14111.pdf The proof is not really constructive, one obtains a sequence of completely labeled subsimplices that serve as "almost-fixed-points", which may not be close to any actual fixed point. This sequence is in general not convergent, but one gets a convergent subsequence using compactness and this subsequence converges to an actual fixe-point. $\endgroup$
    – Michael Greinecker
    Commented Sep 24, 2012 at 16:33
  • $\begingroup$ @Michael Unfortunately, I cannot read German. $\endgroup$
    – Liviu Nicolaescu
    Commented Sep 24, 2012 at 17:42
  • $\begingroup$ There was a brief vogue for using this approach to solve fixed-point problems in economics. Herb Scarf wrote an elementary introduction that's available at cowles.econ.yale.edu/~hes/pub/fixed%20point%20theorems.pdf $\endgroup$
    – arsmath
    Commented Sep 25, 2012 at 15:56
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In these notes by Tony Carbery, it is mentioned that a proof along these lines appears in the book Differential Forms and Applications by do Carmo, where it is attributed to E. Lima.

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Could one of these two be what you're looking for?

  1. J. Milnor, Analytic proofs of the “hairy ball theorem” and the Brouwer fixed-point theorem, Amer. Math. Monthly 85 (1978), no. 7, 521–524. MR MR505523 (80m:55001)
  2. C. A. Rogers, A less strange version of Milnor’s proof of Brouwer’s fixed-point theorem, Amer. Math. Monthly 87 (1980), no. 7, 525–527. MR MR600910 (82b:55004)
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According to Lax in his expository paper Change of Variables in Multiple Integrals, Hadamard's original proof of the Brouwer fixed-point theorem from 1910 is based on the determinant formula for change of variables in integrals. The reference he gives is

  • J. Hadamard, Sur quelques applications de l'indice Kronecker, pp. 437-477, in J. Tannery, Introduction a la theorie des functions d'une variable, vol. 2, Paris, 1910.
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Here are two further references of proofs of the fixed point theorem that rely on evaluating determinants:

MR0117523 Dunford, Nelson ; Schwartz, Jacob T. Linear Operators. I. General Theory. With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London 1958 xiv+858 pp. The proof is on page 467.

MR0610487 Kannai, Yakar . An elementary proof of the no-retraction theorem. Amer. Math. Monthly 88 (1981), no. 4, 264--268.

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  • $\begingroup$ Whoh! As a reference for a "simple" proof, he gives an 800-page book! Please, can someone perhaps provide a sub-interval of the pages... $\endgroup$
    – Gerald Edgar
    Commented Sep 24, 2012 at 14:08
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    $\begingroup$ @GeraldEdgar, the books has an index with the entry "Brouwer fixed point theorem, proof of, (467)". The proof ends on the top of page 470. $\endgroup$
    – Michael Greinecker
    Commented Sep 24, 2012 at 14:32
  • $\begingroup$ @Michael: There is no harm in including the page ref -- I agree with @Gerald, even if this is not as hard to look up as it could be... $\endgroup$
    – Igor Rivin
    Commented Sep 24, 2012 at 20:05
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There is an interesting essay on Brouwer's Fixed Point theorem, including a contructive proof, at Kevin Brown's MathPages site http://www.mathpages.com/home/kmath262/kmath262.htm

Looking at the home page, I see he has now written a string of books. If his articles are any indication, these books are doubtless excellent and well worth buying.

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