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

Abstract

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".

A plasticity principle of some generalized Gauss trees

“…We shall consider the same set of variables for the length of linear segments and angles that have been used in [16][17][18]. Let 1 2 3 4 5 6 7 8 be a heptagonal pyramid (closed octahedron) as shown in Figure 1 with the convex heptagon 1 2 3 5 6 7 8 as a base and the point 4 in ℝ 3 as a vertex.…”

“…We shall prove that the objective function in (2.1) depends on 1 , 2 , 8 , 3 , 5 and 6 . Applying the same method that was used in [15][16][17][18], we express 4 and 00 ὔ as functions of 1 , 2 and 8 with respect to 1 2 3 4 0 ὔ 8 , by eliminating the dihedral angle .…”

“…We denote by ( ) 12840 ὔ the weight which corresponds to the vertex that lies on the ray 0 for = 1, and Replacing (3.8) and (3.9) in (2.13), we deduce that Taking into account the solution of the 4-inverse weighted Fermat-Torricelli problem (see [15,17]) we calculate ( / ) with respect to : 2 = sin 2 ( 0 ) − cos 2 ( 0 ) − cos 2 ( 0 ) + 2 cos( 0 ) cos( 0 ) cos( 0 ) sin 2 ( 0 ) − cos 2 ( 0 ) − cos 2 ( 0 ) + 2 cos( 0 ) cos( 0 ) cos( 0 ) The next theorem gives a plasticity principle of a generalized Gauss tree of degree at most ve with respect to given vertices which form a heptagonal pyramid in ℝ 3 (see Figure 1). …”

Weighted area-minimizing submanifolds with soap-film-like singularities assembled from special Lagrangian pieces

“…The constructed examples are also fit for a more general setting when the volume of each piece S i is weighted by a positive real number λ i (see Theorem 1) as in the study of immiscible fluids with interface energies [9, Section 2], clusters with multiplicities [10], cost-minimizing networks [5], or in the weighted Fermat-Torricelli problem [6,12], which can be viewed as the one-dimensional case of WAMD in R 2 .…”

Το πρόβλημα Fermat-Torricelli και ένα αντίστροφο πρόβλημα στο Κ-επίπεδο και σε κλειστά πολύεδρα του R^3

“…Τα αποτελέσματα του Κεφαλαίου αυτού είναι πρωτότυπα κι έχουν δημοσιευτεί στις εργασίες [88] (τετράεδρα), [89] (κλειστά πεντάεδρα) και [91] (κλειστά εξάεδρα).…”