Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron We know 'triangle inequality'. I'm interested in the generalization of this inequality.
Here is my question.
Question: How can we represent a necessary and sufficient condition for each positive number $a, b, c, d$ being each area of four faces of a tetrahedron?
I've tried to get some kind of inequality, but I'm facing difficulty.

$$a<b+c+d$$
$$b<c+d+a$$
$$c<d+a+b$$
$$d<a+b+c$$
Is that what you want?
This condition is necessary. You can't have sufficient condition with areas alone.
Let's investigate if a tetrahedron with triangle areas $a,b,c,d$ is possible if the conditions are true.
Let $a>b, a>c, a>d$. Make $a$ a regular triangle with side $A$. Now you must construct one more point on position $(x,y,z)$. Triangles $b,c,d$ must have heights $H_b,H_c,H_d$ such that $AH_b/2=b$ etc. Now we have three constraints and three variables, can you solve it?
