Determine the tetrahedron $ABCD$ so that $\frac{L}{R}$ has the maximum value, where $L=AB+AC+AD+BC+BD+CD$, and $R$ is the radius of the circumsphere of $ABCD$. sorry about my bad English ^-^
Determine tetrahedron maximizing sum of edges over radius of circumsphere
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$\begingroup$
geometry
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0Can you find a formula for $R$ in terms of side lengths and angles? – 2012-04-03
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1It seems "obvious" that the tetrahedron you want is regular. If so, formulas for the radius of the sphere are available on the web. – 2012-04-04
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1The regular tetrahedron has $L/R\doteq 9.8$, a flat square degenerate tetrahedron has $L/R\doteq9.65$. – 2013-02-08
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0@RossMillikan It is natural to think that the tetrahedron maximizing the given isoperimeter is probably regular. But do you have any clue how to prove it? – 2015-02-14