Abstract

We obtain a generalized Fermat-Torricelli tree for a boundary pentagonal pyramid (closed hexahedron) and a boundary closed enniaedron in the three dimensional Euclidean Space by placing two mobile vertices with positive weights at the interior of their convex hull and by introducing a method of differentiation of the objective function with respect to two variable dihedral angles and four variable length of linear segments which connect each mobile vertex with two boundary vertices. The generalized Fermat-Torricelli tree with degree at most four with respect to a closed hexahedron or a closed enniaedron is an Euclidean minimal tree structure which may be considered as an intermediate class of a tree structure between a weighted Fermat-Torricelli (tree) structure and a weighted Steiner minimal tree structure with respect to a closed hexahedron or closed enniaedron.

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

  • Generalized Fermat-Torricelli tree
  • generalized Fermat-Torricelli constant
  • pyramids
  • closed polyhedra

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. K. Bopp: Über das Kürzeste Verbindungssystem zwischen vier Punkten, Dissertation, Göttingen (1879).Google Scholar
  4. D. Cieslik: Steiner Minimal Trees, Nonconvex Optimization and its Applications 23, Kluwer, Dordrecht (1998).https://doi.org/10.1007/978-1-4757-6585-4Google Scholar
  5. A. Cotsiolis, A. Zachos: Corrigendum to "The weighted Fermat-Torricelli problem on a surface and an "inverse" problem", J. Math. Anal. Appl. 376(2) (2011) 760.https://doi.org/10.1016/j.jmaa.2010.12.021Google Scholar
  6. R. Courant, H. Robbins: What is Mathematics?, Oxford University Press, New York (1951).Google Scholar
  7. E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29.https://doi.org/10.1137/0116001Google Scholar
  8. V. Jarnik, R. Kossler: O minimalnich grafeth obeahujicich n danijch bodu, Cas. Pest. Mat. a Fys. 63 (1934) 223–235.https://doi.org/10.21136/CPMF.1934.122548Google Scholar
  9. A. O. Ivanov, A. A. Tuzhilin: Minimal Networks, the Steiner Problem and its Generalizations, CRC Press, Boca Raton (1994).Google Scholar
  10. A. O. Ivanov, A. A. Tuzhilin: Weighted minimal binary trees, Uspekhi Mat. Nauk 50(3) (1995) 155–156 (in Russian); Russian Math. Surv. 50(3) (1995) 623–624 (in English).https://doi.org/10.1070/RM1995v050n03ABEH002569Google Scholar
  11. V. A. Toponogov: Differential Geometry of Curves and Surfaces, Birkhäuser, Basel (2005).Google Scholar
  12. 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
  13. 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
  14. A. Zachos, A. Cotsiolis: The weighted Fermat-Torricelli problem on a surface and an "inverse" problem, J. Math. Anal. Appl. 373(1) (2011) 44–58.https://doi.org/10.1016/j.jmaa.2010.06.029Google Scholar
  15. A. Zachos, G. Zouzoulas: An evolutionary structure of convex quadrilaterals. Part II, J. Convex Analysis 20(2) (2013) 483–493.https://doi.org/10.68381/jca20029Google Scholar
  16. A. Zachos: A weighted Steiner minimal tree for convex quadrilaterals on the two-dimensional K-plane, J. Convex Anal. 18(1) (2011) 139–152.https://doi.org/10.68381/jca18007Google Scholar
  17. A. N. 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

Suggested citation

A. Zachos. “A Generalized Fermat-Torricelli Tree in the Three Dimensional Euclidean Space.” Journal of Convex Analysis 24 (2017), No. 1, 185–198. https://doi.org/10.68381/jca24014

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2017. Rights now held by Banach Press.