Edge lengths determining tetrahedrons

For a tetrahedron to actually exist, the edge lengths of each of its four faces evidently must obey the triangle inequality. This condition is necessary but not sufficent for six edge lengths to make up a tetrahedron. There does, for example, not exist a tetrahedron with five edges of length 4 and one edge of length 7, even though the triangle inequalities are fulfilled. Just consider two equilaterals with edge length 4 as faces of a tetrahedron; the remaining edge length must be smaller than 4 √ 3 (< 6.93), since this is the extreme value reached when the tetrahedron becomes degenerate (see Fig. 1). When are six given lengths the edge lengths of some tetrahedron? This question has been addressed to already several times in the literature (Menger, Blumenthal, Dekster and Wilker, Herzog, see below), mostly even for the general case of d-dimensional simplices. The present work, as an offshoot of our original investigations concerning tetrahedral structures in organic chemistry, is restricted to the 3-dimensional case. This restriction