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

Abstract

We introduce the weighted Fermat-Torricelli-Menger problem for a given sextuple of edge lengths in R 3 which states that: given a sextuple of edge lengths determining tetrahedra and a positive real number (weight) which corresponds to each vertex of every derived tetrahedron find the corresponding weighted Fermat-Torricelli point of these tetrahedra. We obtain a system of three rational equations with respect to three variable distances from the weighted Fermat-Torricelli point to the three vertices of the tetrahedron determined by a given sextuple of edge lengths in the sense of Menger. This system of equations gives a necessary condition to locate the weighted Fermat-Torricelli point at the interior of this class of tetrahedra and allow us to compute the position of the corresponding weighted Fermat-Torricelli point. Furthermore, we give an analytical solution for the weighted Fermat-Torricelli-Menger problem for a given sextuple of equal edge lengths in R 3 for the case of two pairs of equal weights.

Isometric embedding of a weighted Fermat-Frechet multitree for isoperimetric deformations of the boundary of a simplex to a Frechet multisimplex in the $K$-Space

“…In [53], we found the position of the weighted Fermat-Torricelli trees for the weighted Fermat-Frechet problem for a given sextuple of positive real numbers determining the edge lengths of tetrahedra in R 3 , by substituting Caley-Menger determinants in some weighted volume entropy equalities for tetrahedra derived in [49].…”