Everyone knows Heron formula for the area of a triangle as a function of its sides. Moving to three dimensions, what is the maximum volume of a tetrahedron with given four face areas $a$, $b$, $c$, and $d$?
Obviously, this is an optimization problem. For example, with $a=b=c=d=1$ one can construct a tetrahedron with any volume between zero and $\frac{2\sqrt2}{3\sqrt{3\sqrt3}}$. The latter is the volume of the regular tetrahedron with side 1, and it is this tetrahedron that gives the maximum volume (from symmetry considerations). The zero volume is presented by any degenerate tetrahedron (such as the "tetrahedrons" with vertices given by$$A=(0,0,0),\quad B=(2,0,0),\quad C=(1+e,1,0),\quad D=(1-e,-1,0),$$for any real $e$, all have four faces 1 and volume zero).
A degenerate tetrahedron can not be constructed for given face areas when $S_1\not=S_2+S_3+S_4$ and $S_1+S_2\not=S_3+S_4$ for any assignment of $a,b,c,d$ to $S_1,S_2,S_3,S_4$. Then a complementary problem is what is the minimum volume of a tetrahedron with given four faces $a$, $b$, $c$, and $d$?
The "maximum" problem somewhat resembles "What is the maximum area of the quadrilateral given by its sides $a$, $b$, $c$, and $d$?", for which Brahmagupta's formula gives the solution. See also this question, and you can find numerous other pieces of study of this planar problem in the Internet. In this problem one can intuitively guess that the four vertices of the maximum area quadrilateral should live on a circle, and then proceed with the derivation and proof of the formula. For the maximum volume of tetrahedron, what intuitive guess can be made?
I suspect the maximum volume is obtained together with the maximum sum of the six digedral angles at the six edges of the tetrahedron. In the above example, the sum is $2\pi\approx6.28$ for the degenerate tetrahedra and $6\cdot2\arcsin\frac1{\sqrt3}\approx7.39$ for the regular tetrahedron. To support this intuition, the volume is given by any of the six identities like this one:$$3V\cdot AB=2c d\sin AB_\angle,$$where $c$ is the area of the face opposing vertex $C$, and $AB_\angle$ is the dihedral angle at edge $AB$, so all the dihedral angles should be maximized.
From this guess on, the radius of the circumscribed sphere should be minimized, perhaps. I can not prove this guess, but it concords with the expression of the area of a triangle given by its sides: $S=4abc/R$ (if finding the area of the triangle by its sides were an optimization problem, then you would minimize the circumcircle).
The problem was originally given to me by Ernst D. Krupnikov, who claims its closed form solution is cumbersome and difficult to derive without a help of a computer algebra system. Here is a particular case of the problem you may find amusing to solve: what is the maximum volume of the tetrahedron with face areas 120, 150, 160, 250?
