
Journal of Convex Analysis 27 (2020), No. 3, 9891002 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 FermatTorricelli problem for tetrahedra in R^{3} by assigning to the corresponding weighted FermatTorricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted FermatTorricelli trees of degree five for boundary closed hexahedra in R^{3} by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R^{3}. We also derive a complete solution for an important generalization of the inverse weighted FermatTorricelli problem for three noncollinear points and a new plasticity principle of mass networks of degree four for boundary convex quadrilaterals in R^{2}. 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: FermatTorricelli problem, inverse FermatTorricelli problem, tetrahedra, plasticity of closed hexahedra, plasticity of quadrilaterals. MSC: 51E10, 52A15, 52B10. [ Fulltextpdf (122 KB)] for subscribers only. 