Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
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asked Aug 28, 2012 at 14:11
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$\begingroup$ Please add the open-problem tag to this question. To my knowledge only a couple of the conjectures mentioned in the linked survey have been resolved. (For example, Denis Serre asked almost 2 yrs ago the following question: mathoverflow.net/questions/42902/… which is still an open problem (a semi-famous one at that). $\endgroup$– SuvritAug 28, 2012 at 14:30
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7$\begingroup$ No offence intended, but this question comes off really lazyily formulated. Couldn't you state or at least parphrase the conjecture here, and only in addition link to the document? Some people (me sometimes, but not at the moment) read the site with mobile devices, it then is not very convenient (sometimes even impossible) to open some pdf and browse around it. Also, to say it is Problem 7 is a bit missleading and costed me some aditional time in finiding it. It seems to be Conjecture 11 in Section 7 (okay, reading the intro it becomes clear, but going there directly is harder). $\endgroup$– user9072Aug 28, 2012 at 14:40
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$\begingroup$ Well, sorry about the trouble I gave you. I'll bear that in mind next time, promise! $\endgroup$– Felix GoldbergAug 28, 2012 at 14:47
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$\begingroup$ Thanks for the reply; and just in case you wonder, the vote to close is not from me (it was already around before I commented). $\endgroup$– user9072Aug 28, 2012 at 14:58
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$\begingroup$ Didn't suspect it was from you actually, but awfully kind of you to disassociate from it, nevertheless. :) $\endgroup$– Felix GoldbergAug 28, 2012 at 16:24
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2 Answers
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To my knowledge, this problem is still open. There has been partial progress on it in the past few years, but it still seems quite far from resolved.
The latest paper that I am aware of is: here, though unfortunately behind a pay wall.
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$\begingroup$ That latest paper is really silly. Using Cauchy-Schwarz inequality, one may produce a mild condition. $\endgroup$– BetrandSep 6, 2012 at 1:05
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$\begingroup$ @Betrand: interesting. I just saw the title of the paper, but did not read it! $\endgroup$– SuvritSep 6, 2012 at 8:08
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$\begingroup$ @Bertrand: I don't quite understand what you mean. Could you please explain? $\endgroup$ Sep 10, 2012 at 11:10
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Here is a more recent paper: Roman Drnovšek, On the S-matrix conjecture. It (re-) proves for the case of $n$ odd the conjecture restated here for convenience:
If $A$ is a nonsingular $n\times n$ matrix with all entries in the interval $[0,1]$, then
$$||A^{-1}||_F\ge\frac{2n}{n+1}$$ with equality iff $A$ is an $S$-matrix.
