The weighted Fermat–Torricelli problem for tetrahedra and an “inverse” problem

We study the weighted Fermat–Torricelli problem for tetrahedra in R 3 and solve an “inverse” problem by introducing a method of differentiation. The solution of the inverse problem is the main result which states that: Given the Fermat–Torricelli point A 0 with the vertices lie on four prescribed rays, find the ratios between every pair of non-negative weights of two corresponding rays such that the sum of the four non-negative weights is a constant number. An application of the inverse weighted Fermat–Torricelli problem is the strong invariance principle of the weighted Fermat–Torricelli point which gives some classes of tetrahedra that could be named “evolutionary tetrahedra”.