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Jung's theorem states that every compact set $K \subset \Bbb{R}^n$ of diameter $d$ is contained in some closed ball of radius $$ r \leq d \sqrt{\frac{n}{2(n+1)}} $$ with equality attained for the regular $n$-simplex of side $d$.

Where can I find a proof of this result? The relevant page on Wikipedia mentions the original articles, which would be fine for me if they were in English, while none of the other references has a proof (at least not in full generality).

A discussion of (some of) its generalizations would be a nice bonus, too, but it isn't required.

A.P.
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  • I am made aware of the theorem, for me which is beautiful, by your question. Consider adding a tag such as "reference request" :). – Yes Sep 02 '16 at 20:02
  • @Nobody Thank you for mentioning the [tag:reference-request] tag... I totally forgot about it. I'm glad you like this theorem, as it is indeed quite elegant! – A.P. Sep 02 '16 at 20:20
  • For instance in: https://matthewhr.wordpress.com/2013/03/14/hellys-theorem-and-applications-ii-jungs-theorem/ (google is your friend, right?). – Moishe Kohan Sep 03 '16 at 16:27
  • @studiosus DuckDuckGo — which is my main search engine — shows (almost) only results about another theorem of Jung's, better known as the Abhyankar-Jung Theorem, for the first few pages searching for "jung theorem proof". Similarly, every result in the first three pages seems to be about that other theorem, but for one which is a proof on Cut the Knot for $n = 2$... Apparently Google isn't my friend, but it's definitely yours... – A.P. Sep 03 '16 at 18:11

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A proof can be found for instance at https://matthewhr.wordpress.com/2013/03/14/hellys-theorem-and-applications-ii-jungs-theorem/, a sketch of the proof can be found in HTFB's answer to this MSE question.

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Moishe Kohan
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