An $\epsilon$ characterization of the vertices of tetrahedra in the three dimensional Euclidean Space

Abstract

We determine a positive real number (weight), which corresponds to a vertex of a tetrahedron and it depends on the three weights which correspond to the other three vertices and an infinitesimal number $\epsilon.$ As a limiting case, for $\epsilon\to 0,$ the quad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with the three neighbouring edges of the tetrahedron and intersect at this vertex.