Talk:Kakutani fixed-point theorem

Infinite dimensions

At the end of Fixed point theorems in infinite-dimensional spaces a much more general version of this theorem is stated:

Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.

I don't know if it's true and no reference is given.AxelBoldt 04:32, 12 June 2006 (UTC)

This is true. So I have added it with a reference. Jyotirmoyb 03:59, 30 July 2007 (UTC)

Yes, it's more or less true, but the statement is a bit garbled. Here's a cleaner version:

Let V be a topological vector space and T : V → V a continuous affine map. If K is a compact convex subset of V such that TK ⊂ K, then Tp=p for some p ∈ K.

More generally, if F is a collection of mutually commuting continuous affine maps, such that TK⊂K for all T ∈ F, then the collection F has a common fixed point in K. Alternatively, the conclusion also holds if F is an equicontinuous group of affine maps. Silly rabbit 16:06, 15 June 2006 (UTC)

It just occurred to me that the version you quoted is for correspondences, not for self-maps of the space. So I don't know any natural way to relate the two versions of the theorem, so I have no idea either. Silly rabbit 21:04, 15 June 2006 (UTC)

I think that j should be just called a function and not a correspondence. It is a function to 2^S and corrspondence to S. Right? 15:19, 19 August 2006 (UTC)

Convexity assumptions

I just reinserted the assumption that S be convex; without it, the theorem is clearly false (by taking S=S¹ and j(x) to be singletons, rotating the circle). The statement in the Encyclopaedia of Mathematics is therefore wrong and I removed the reference and reinstated the older one.

I'm not sure whether we really need to assume that j(x) be convex for all x. That's why I added the {{verify}} tag. AxelBoldt 08:05, 13 June 2006 (UTC)

Convexity of j(x) is indeed necessary. Consider $f(x):[0,1]\rightarrow 2^{[0,1]}$ where

$f(x)={\begin{cases}1&0\leq x<0.5\\\{0,1\}&x=0.5\\0&0.5<x\leq 1\\\end{cases}}$

$f(x)$ satisfies all other conditions of the theorem but still does not have a fixed point. Therefore, if there are no other objections I will remove the verify tag in a few days. Jyotirmoyb 10:04, 28 July 2007 (UTC)

There seems to be an extension ([1]) of the theorem that does not require convexity of j(x) at the boundaries of S. But the result is available only in a working paper and not in any published source. Should we include it in the article? Jyotirmoyb 12:09, 28 July 2007 (UTC)

Link for closed graph?

Right now the closed graph condition on the function links to closed graph theorem with which we only share the definition of a closed graph and the rest of the content may easily confuse a reader. On the other hand graph of a function mixes up the set-theoretic notion with diagrammatic representations. Any suggestions for a better link?

For now, have replaced the link with a footnote which defines a closed graph. This is not neat, but I couldn't think of anything else short of creating a new article on the set-theoretic concept of the graph of the function. Jyotirmoyb 03:39, 30 July 2007 (UTC)

Prelim comments

Some issues

• Proof or outline thereof should be explained.
• Equivalence of the two forms should be shown I think.
• I think accessibility to the main public is a problem. I know what the Brouwer FPT is but most people who have not reached 3rd or fourth year pure maths would have no idea. Some background on topology and what topology is is defnitely needed such what a FPT is.
• Need more pure maths facets also. ATM we only have RL applied applications. Need to show how this links into other toplogical stuff, so that pure maths people can get an idea of its significance also. Is there any hisotry about the events leading up to the discovery of the theorem? IS it used for further pure maths developments?
• Lead is too short and it is supposed to tell us what summary of the article is. At the moment the first sentence is not expanded on in the main body, leaving basically only one line.

Blnguyen (bananabucket) 09:34, 7 August 2007 (UTC)

Thanks for the comments. I shall be addressing them in the next couple of days. Jyotirmoyb 04:43, 8 August 2007 (UTC)

• Gave argument for the equivalence of the two forms. Jyotirmoyb 04:43, 8 August 2007 (UTC)
• Expanded lead and made it independent of prior-knowledge of Brouwer's theorem. Jyotirmoyb 05:37, 8 August 2007 (UTC)
• Added proof outline. Jyotirmoyb 06:00, 9 August 2007 (UTC)
• I think we should not add too many details on topology or fixed point theorems to this article both because given the nature of the topic it is reasonable to expect readers to have some mathematical background and because that material would be common to all topological fixed point theorems and rightly belongs to 'Fixed point theorem'.
• It would be good to have more material on historical background and links with other mathematical results apart from Brouwer's theorem. But I don't have the expertise or the resources to take this up right away. It will have to wait for other editors.

I will stop this iteration of editing now and wait for further comments from the GA reviewers and other editors. Jyotirmoyb 08:27, 10 August 2007 (UTC)

Can you put refs for all paragraphs? Blnguyen (bananabucket) 08:41, 15 August 2007 (UTC)

I had to fail this for the time being becuase the author is away. Blnguyen (bananabucket) 01:49, 17 August 2007 (UTC)

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Mistake in section "Relation to Brouwer's fixed-point theorem"

It invokes the "approximate selection theorem" in the proof of Kakutani's Fixed-Point Theorem, but the conditions of the Kakutani Fixed-Point Theorem don't imply the conditions of the "approximate selection theorem". In particular, one of the seemingly very strong conditions is not implied:

> For every $\varepsilon >0$ there exists a continuous function $f:X\rightarrow Y$ with $\operatorname {graph} (f)\subset [\operatorname {graph} (F)]_{\varepsilon }$, where $[S]_{\epsilon }$ is the $\epsilon$-dilation of $S$, that is, the union of radius-$\epsilon$ open balls centered on points in $S$.

So the approximate selection theorem can't be used. The "proof" is wrong. I might add a banner. --Svennik (talk) 12:26, 19 December 2023 (UTC)

Annoyingly, it was actually the article on Selection theorem which was wrong. The mistaken statement of the approximate selection theorem is from December 2022, which I then corrected. There is nothing wrong with the proof in the article on Kakutani's theorem. Reverting my banner. --Svennik (talk) 13:51, 19 December 2023 (UTC)