Timeline for Minimum diameter of set inscribed in a unit sphere
Current License: CC BY-SA 4.0
13 events
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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 12:06 | comment | added | Ivan Izmestiev | I see. I edited the answer. But anyway, BerndM found the right reference. | |
| Jan 13, 2023 at 12:05 | history | edited | Ivan Izmestiev | CC BY-SA 4.0 |
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| Jan 13, 2023 at 11:31 | comment | added | Roland Bacher | I think you should not use $n$ in Caratheodory's theorem: It is perhaps a smaller $n$ than in the original question. | |
| Jan 13, 2023 at 10:57 | comment | added | Roland Bacher | The intersection of $A$ with the sphere can be a simplex of small dimension but $A$ is not necessarily contained in the affine hull of this simplex. Therefore the is perhaps a snag with applying Caratheodory directly. | |
| Jan 13, 2023 at 10:46 | comment | added | Ivan Izmestiev | I do not understand your question. If you mean the induction base, then it is $n=1$, both the sphere and the simplex are segments, the lower bound is 2. | |
| Jan 13, 2023 at 10:41 | comment | added | Roland Bacher | OK. Where do you treat the problem of $A$ reduced to $2$ points (and similar stuff given by large simplices of lower dimension) | |
| Jan 13, 2023 at 10:41 | history | edited | Ivan Izmestiev | CC BY-SA 4.0 |
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| Jan 13, 2023 at 10:38 | comment | added | Ivan Izmestiev | I assumed that the new simplex also contains the center of the sphere. Will add this to my answer. | |
| Jan 13, 2023 at 10:33 | comment | added | Roland Bacher | The statement 'then the opposite face is closer to the center ...' is not true assuming only that the simplex is inscribed in a sphere: A very tiny simplex which is almost flat can be inscribed in a huge sphere and all its faces can be far away from the origin. One must therefore also use the assumptions. | |
| Jan 13, 2023 at 8:41 | history | edited | Ivan Izmestiev | CC BY-SA 4.0 |
Edit: for a more general statement on edge lengths of inscribed simplices see Ilya Bogdanov's answer to [this question][1]:; added 125 characters in body
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| Jan 12, 2023 at 13:16 | history | edited | Ivan Izmestiev | CC BY-SA 4.0 |
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| Jan 12, 2023 at 13:04 | history | answered | Ivan Izmestiev | CC BY-SA 4.0 |