# Reference request: an elementary proof of Brouwer fixed-point theorem.

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.

• I think its come from DUnford Schwartz Linear OPerators I p. 467 – Buschi Sergio Sep 25 '12 at 12:53

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.

• It answers the title of the question. :) – Jim Conant Sep 24 '12 at 13:25
• 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. – Michael Greinecker Sep 24 '12 at 16:33
• @Michael Unfortunately, I cannot read German. – Liviu Nicolaescu Sep 24 '12 at 17:42
• 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 – arsmath Sep 25 '12 at 15:56

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.

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)

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.

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.

• 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... – Gerald Edgar Sep 24 '12 at 14:08
• @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. – Michael Greinecker Sep 24 '12 at 14:32
• @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... – Igor Rivin Sep 24 '12 at 20:05

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.