The weighted Fermat-Torricelli problem for tetrahedra and an ``inverse'' problem
Let \(A_1\), \(A_2\), \(A_3\), \(A_4\) be four affinely independent points in \(\mathbb{R}^3\) with corresponding positive weights in each case. The authors ask for the weighted Fermat-Torricelli point \(A_1\) of that point system, i.e., for the unique and existing point \(A_0\) minimizing the sum of weighted distances to the four given points. They propose a new method and solve an ``inverse'' Fermat-Torricelli problem by differentiating some geometric relations for tetrahedra with respect to specific dihedral angles. This reads as follows: Given \(A_0\) with the four points from above on prescribed rays, find the ratios between every pair of nonnegative weights of two corresponding rays such that the sum of the four nonnegative weights is a constant number. The authors point out that this inverse problem cannot be solved via the usual numerical approach, also known as ``Weiszfeld algorithm''. An application of their results is the introduction of some specific classes of tetrahedra.
scientific article; zbMATH DE number 5549409