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Jan 16, 2023 at 17:54 comment added Steve In a comment, BerndM informed me that the result I'm looking for is "Jung's theorem". See Wikipedia. It turns out that Jung's proof is rather long, so maybe this problem is harder than it looks. For the case of a circle (n=1) see the reference to Rademacher and Toeplitz given in the Wikipedia article. Even that proof is rather involved.
Jan 13, 2023 at 11:29 comment added Roland Bacher I rechecked it and I did not see problems but perhaps I am blind to an obvious mistake.
Jan 12, 2023 at 15:51 comment added Steve I don't find this argument convincing.
Jan 10, 2023 at 17:12 history answered Roland Bacher CC BY-SA 4.0