The Fermat-Torricelli point and isosceles tetrahedra

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

“…For extreme cases, we refer to [3] and [2] (absorbed case of Theorem 1). Special cases are tetrahedra with equal non-negative weights and regular tetrahedra with unequal non-negative weights.…”

“…A unified approach of the weighted Fermat-Torricelli problem in the plane, two-dimensional sphere and two-dimensional hyperboloid is given in [5]. For a unified approach to geometric properties of the Fermat-Torricelli point of four affinely independent points, we refer to [3] and for n affinely independent points in [2]. In this paper, we provide a new method, in order to study the weighted Fermat-Torricelli problem for tetrahedra in R 3 and solve an "inverse" problem by differentiating some geometric relations for tetrahedra with respect to specific dihedral angles.…”

The weighted Fermat–Torricelli–Menger problem for a given sextuple of edge lengths determining tetrahedra

“…The existence and uniqueness of the weighted Fermat-Torricelli point and a complete characterization of the solution of the weighted Fermat-Torricelli problem for tetrahedra has been established in [5, Theorem 1.1, Reformulation 1.2 page 58, Theorem 8.5 page 76, 77], [8] and for specific cases in [3,4].…”

The product of the distances of a point inside a regular polytope to its vertices

“…A classical case of great interest is the Fermat-Steiner problem of minimizing the sum of the distances PP 1 + PP 2 + PP 3 for a variable point P in the plane of a given triangle P 1 P 2 P 3 ; the recent paper of Gueron and Tessler [3] gives an interesting discussion of the history and variants of this problem. The paper of Kupitz and Martini [4], dealing with the analogous problem in E 3 , provides a useful supplement to the references in [3].…”