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Sep 15, 2010 at 3:00 comment added Nikita Wow, that's a pretty serious body of work! I'm going to hold off marking this as the accepted answer for a while in the hope that someone will suggest an easier proof of the fact that I asked for (which is much weaker than these papers prove!).
Sep 15, 2010 at 2:54 comment added Bill Johnson Start with projecteuclid.org/DPubS/Repository/1.0/… Also look for Bessaga's papers, maybe joint with Klee, in the 1960s in MathSciNet (which I cannot access right now).
Sep 15, 2010 at 2:32 comment added Nikita Do you know a reference for this?
Sep 15, 2010 at 2:30 history answered Bill Johnson CC BY-SA 2.5