Talk:Kakutani fixed-point theorem

message: 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  tag. AxelBoldt 08:05, 13 June 2006 (UTC)
replies: text: There seems to be an extension () 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)
thread_title: Convexity assumptions

message: 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)
thread_title: Link for closed graph?

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thread_title:  External links modified 

message: 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)
replies: text: 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)
thread_title:  Mistake in section "[[Kakutani_fixed-point_theorem#Relation_to_Brouwer's_fixed-point_theorem|Relation to Brouwer's fixed-point theorem]]"