Status of the compact AR problem?

2011-09-20T00:02:46Z

The so-called "compact AR Problem" reads:

Is every compact convex set in a metrizable topological vector space an absolute retract?

It is open according to the chapter by T. Banakh, R. Cauty and M. Zarichnyi in "Open Problems in Topology II" (2007) and according to a 2009 paper, which acknowledges discussions of its authors with Nguyen To Nhu.

Yet according to Sehie Park's 2004 survey and another 2009 paper it has been resolved affirmatively in a 8-page paper by Park published in some conference proceedings in 2004. (If you happen to solve "one of the most outstanding open problems in infinite dimensional topology", apparently you may have an urge to ignore top international journals and submit it to Antarctica Journal of Mathematics - especially if you happen to live in Antarctica.)

Given that Cauty and Nhu have been long active in the area and have obtained substantial results related to the problem, it would be strange if they missed a correct solution. But it is also strange that they didn't mention Park's paper (whatever they think of it).

So what is going on here?

(I must admit that so far I have mostly checked out just those sources that are freely accessible on the internet, but given that some confusion already exists in at least one of the two 2009 papers, I thought that even if I'm missing a trivial solution such as MathSciNet it is perhaps already legitimate to ask this question here, so as to make that trivial solution more widely known.)

Answer by Matthew Daws for Status of the compact AR problem?

2011-09-20T13:10:50Z

So if I understand right, you haven't checked MathSciNet? Well, it's not much use:

Park's paper is http://www.ams.org/mathscinet-getitem?mr=2144066 but there is no review.
Park's survey paper is http://www.ams.org/mathscinet-getitem?mr=2188170 but there is only a "summary".
Those are the only things you are lead to by citations.

(Community Wiki so others can edit if they wish; and this is hardly an answer worthy of gain rep for.)

Edit (Sergey Melikhov): As I finally got the chance to get to my office and browse Mathscinet, I am led to something interesting by citations. The 2004 paper of Park, which has no review, is cited in the review of his another paper [Remarks on recent results in analytical fixed point theory. Nonlinear analysis and convex analysis, 517–525, Yokohama Publ., Yokohama, 2007], which says:

Summary: "We show that some fixed point theorems and related results in our previous works [a long list of Park's papers including the 2004 paper] need additional requirements for their validities. Some of the new correct results appear in [S. Park, J. Nonlinear Convex Anal. 7 (2006), no. 1, 1-17]."

The review of the latter paper doesn't explicitly mention the compact AR problem, but it will probably take me a while to get the 2006 and 2007 papers.

Status of the compact AR problem? - MathOverflow

Status of the compact AR problem?

Answer by Matthew Daws for Status of the compact AR problem?