Journal of Convex Analysis 27 (2020), No. 3, 989--1002
Copyright Heldermann Verlag 2020
The Plasticity of some Mass Transportation Networks in the Three Dimensional Euclidean Space
Anastasios N. Zachos
Dept. of Mathematics, University of Patras, 26500 Rion, Greece
azachos@gmail.com
We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in R3 by assigning to the corresponding weighted Fermat-Torricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted Fermat-Torricelli trees of degree five for boundary closed hexahedra in R3 by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R3. We also derive a complete solution for an important generalization of the inverse weighted Fermat-Torricelli problem for three non-collinear points and a new plasticity principle of mass networks of degree four for boundary convex quadrilaterals in R2. The plasticity of mass transportation networks provides some first evidence for a creation of a new field that we may call Mathematical Botany in the future.
Keywords: Fermat-Torricelli problem, inverse Fermat-Torricelli problem, tetrahedra, plasticity of closed hexahedra, plasticity of quadrilaterals.
MSC: 51E10, 52A15, 52B10.
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