How can triangle inequlity be extended to become tetrahedron inequlity?

Question

Since each side of a tetrahedron is a triangle is it possible to have an inequality that involves triangles that can form a tetrahedron?

See answer to this question, it cover two cases where 1. only the lengths of 6 edges are given and 2. only the areas of 4 faces are given. — achille hui, Dec 18 '13 at 11:32

score 1 · Answer 1 · answered Dec 18 '13 at 11:26

1

If you are asking "Find the necessary and sufficient condition that each of four positive numbers $a,b,c,d$ represents each area of the face of a tetrahedron", then it is known that the answer is $$a\lt b+c+d,b\lt a+c+d, c\lt a+b+d, d\lt a+b+c.$$

answered Dec 18 '13 at 11:26

mathlove

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score 0 · Answer 2 · answered Dec 18 '13 at 22:12

But if you are asking "Find the necessary and sufficient condition that each of six positive numbers represents each edge of a tetrahedron", then it is known that the answer is the following:

all triangle inequalities for the faces must be fulfilled
the Cayley-Menger Determinant must be positive

How can triangle inequlity be extended to become tetrahedron inequlity?

2 Answers2