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Feb 8, 2011 at 15:34 comment added Sergei Ivanov @Bill: This works for $\ell_3$ and the like, and yes, running this argument for non-smooth norms would be difficult. I don't know how Kakutani proved it either.
Feb 8, 2011 at 14:46 comment added Bill Johnson Nice, but I don't see how to reduce the Kakutani theorem to the special case you treat. IIRC (it has been so long that I did not even remember that Kakutani proved it!) the proof is not very difficult, but I can't get his paper and also don't have access to Amir's book here.
Feb 8, 2011 at 6:23 vote accept Ewan Delanoy
Feb 7, 2011 at 20:55 history answered Sergei Ivanov CC BY-SA 2.5