Abstract

We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in R3\mathbb{R}^3 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\mathbb{R}^3 by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R3\mathbb{R}^3. 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\mathbb{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.

Author information

Contact details are reproduced from the original publication and may be historical.

Anastasios N. Zachos

Dept. of Mathematics, University of Patras, 26500 Rion, Greece

azachos@gmail.com

Keywords

  • Fermat-Torricelli problem
  • inverse Fermat-Torricelli problem
  • tetrahedra
  • plasticity of closed hexahedra
  • plasticity of quadrilaterals

Mathematics Subject Classification

51E10, 52A15, 52B10

References

  1. A. D. Alexandrov: Convex Polyhedra, Springer, Berlin (2005).Google Scholar
  2. V. Boltyanski, H. Martini, V. Soltan: Geometric Methods and Optimization Problems, Kluwer, Dordrecht (1999).https://doi.org/10.1007/978-1-4615-5319-9Google Scholar
  3. E. Engelbrecht: Planimetrischer Lehrsatz, Arch. Math. Phys. 60 (1877) 447–448.Google Scholar
  4. E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29.https://doi.org/10.1137/0116001Google Scholar
  5. S. Gueron, R. Tessler: The Fermat-Steiner problem, Amer. Math. Monthly 109 (2002) 443–451.https://doi.org/10.1080/00029890.2002.11919871Google Scholar
  6. A. O. Ivanov, A. A. Tuzhilin: Minimal Networks. The Steiner Problem and its Generalizations, CRC Press, Boca Raton (1994).Google Scholar
  7. Y. S. Kupitz, H. Martini: Geometric aspects of the generalized Fermat-Torricelli problem, Bolyai Soc. Math. Studies 6 (1997) 55–127.Google Scholar
  8. A. Uteshev: Analytical solution for the generalized Fermat-Torricelli problem, Amer. Math. Monthly 121(4) (2014) 318–331.https://doi.org/10.4169/amer.math.monthly.121.04.318Google Scholar
  9. A. N. Zachos, G. Zouzoulas: An evolutionary structure of convex quadrilaterals, J. Convex Analysis 15(2) (2008) 411–426.https://doi.org/10.68381/jca15029Google Scholar
  10. A. Zachos, G. Zouzoulas: The weighted Fermat-Torricelli problem for tetrahedra and an “inverse” problem, J. Math. Anal. Appl. 353 (2009) 114–120.https://doi.org/10.1016/j.jmaa.2008.11.057Google Scholar
  11. A. Zachos, G. Zouzoulas: An evolutionary structure of pyramids in the three dimensional Euclidean Space, J. Convex Analysis 18(3) (2011) 833–853.https://doi.org/10.68381/jca18052Google Scholar
  12. A. Zachos: A plasticity principle of closed hexahedra in the three dimensional Euclidean Space, Acta Appl. Math. 125(1) (2013) 11–26.https://doi.org/10.1007/s10440-012-9778-zGoogle Scholar
  13. A. Zachos: A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129(1) (2014) 81–134.https://doi.org/10.1007/s10440-013-9831-6Google Scholar

Suggested citation

A. N. Zachos. “The Plasticity of some Mass Transportation Networks in the Three Dimensional Euclidean Space.” Journal of Convex Analysis 27 (2020), No. 3, 989–1002. https://doi.org/10.68381/jca27051

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.