Comments on: Brouwer Fixed Point Theorem

By: Dhruv

Dhruv — Mon, 17 Nov 2008 20:22:46 +0000

the links provided are really interesting to bridge the gap between mathematics and economics especially the scarfs papers on FPT.

By: Carnival of Mathematics « Rigorous Trivialities

Carnival of Mathematics « Rigorous Trivialities — Fri, 11 Jul 2008 13:45:26 +0000

[...] from spheres to discs, Walt at Ars Mathematica gives us a post on the Brouwer Fixed-Point Theorem. He mentions Sperner’s Lemma, which gives [...]

By: Walt

Walt — Mon, 07 Jul 2008 03:11:19 +0000

Thanks for the link, John. Su has some very interesting papers.

By: john

john — Thu, 03 Jul 2008 22:13:06 +0000

If the Scharf article caught your fancy, then you should check out some of Francis Su’s papers (http://www.math.hmc.edu/~su/papers.html). He applies combinatorial versions of fixed point theorems, such as Sperner’s lemma, to fair division problems. Also, his exposition is very clear and easy to read.

By: Walt

Walt — Thu, 26 Jun 2008 20:16:12 +0000

I missed that. That’s really disappointing.

By: albert

albert — Thu, 26 Jun 2008 08:17:01 +0000

Actually, while the algorithm is guaranteed to find a completely labeled subsimplex, such a subsimplex does not necessarily contain an exact fixed point. All we can claim is that it contains an approximate fixed point, i.e. a point x such that f(x) is close to x.

Note that Sperner’s Lemma cannot be recursively applied to a completely labeled subsimplex, because Sperner’s Lemma requires that a subdivision of the simplex to be “properly labled”, i.e. there are constraints on the labels of points on the boundary of the simplex, e.g. for a triangle, points along the (1,2) edge have to be labeled either 1 or 2. This is generally not satisfied by a completely labeled subsimplex.

The step from Sperner’s Lemma to existence of fixed point (Brouwer) is non-constructive: as we take finer and finer subdivisions of the entire simplex, the sequence of completely labeled subsimplexes might not converge, but it must contain a convergent subsequence (due to compactness).

By: Mio

Mio — Thu, 19 Jun 2008 15:50:30 +0000

There’s a publicly accessible American Scientist article
http://cowles.econ.yale.edu/~hes/pub/fixed%20point%20theorems.pdf
linked to from Scarf’s publication page, on this same topic.

By: Walt

Walt — Wed, 18 Jun 2008 16:51:39 +0000

Omar: Yes, the proof Scarf gives is the Sperner lemma one. Normally, when I read the statement of Sperner’s lemma, and I think “wow, that sounds drily technical”. For example, I had that reaction reading the Sperner’s lemma Wikipedia page when I linked to it. Scarf just explains it very clearly, and makes it explicitly algorithmic.

By: Omar Antolín Camarena

Omar Antolín Camarena — Wed, 18 Jun 2008 13:41:18 +0000

I’m a bit confused, the proof you describe sounds exactly like the proof via the Sperner Lemma (with is really a lot like the bisection method in numerical analysis). Are you just saying that in previous expositions you were familiar with you found the proof technical but not in Scarf’s exposition?

By: John Cook

John Cook — Wed, 18 Jun 2008 11:20:23 +0000

It’s amazing how often fixed point theorems come up. Every existence theorem for nonlinear PDEs that I know of ultimately relies on a fixed point theorem.