How big is a tetrahedron? Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and
let $sum(T) = a^3 + b^3 + ... + f^3$.  We wish to compare $vol(T)$ with $sum(T)$.  [ IMO (1961 #2 ) handles the case of triangles. ]
Either by approaching it directly or by expressing the volume as a determinant,
invoking Hadamard's bound, and applying the AM-GM inequality, one finds that
$vol(T) < sum(T)/36$.  However, this hardly seems best possible.
Question: What is the sharpest upper bound for $vol(T)$ in terms of $sum(T)$?
Extra: Generalize to $n$ dimensions.

If $a.b.c.d.e.f$ are the edges of a 4-hedron; and   
if $a$ opposes $d$, $b$ opposes $e$, and $c$ opposes $f$; and   
if $\{A,B,C,D,E,F\}=\{a^2,b^2,c^2,d^2,e^2,f^2\}$, respectively:    
    
then $Q=AD(-A+B+C-D+E+F)$
$+BE(A-B+C+D-E+F)$
$+CF(A+B-C+D+E-F)$
$-ABC-AEF-BDF-DEF$; and
volume $=\frac{\sqrt{Q}}{12}$.
