Abstract

[For Part I see the authors, J. Convex Analysis 15 (2008) 411-426.] We solve explicitly the generalized Gauss problem for convex quadrilaterals in the two dimensional Euclidean Space. By introducing the variable c=cG+∣B1−B4∣+∣B2−B3∣2,wherecG=12c=c_{G}+\frac{|B_{1}-B_{4}|+|B_{2}-B_{3}|}{2}, \quad\text{where}\quad c_{G}=\frac{1}{2} is the Gauss constant and BiB_{i} are positive real variables, such that ∑i=14Bi=1,\sum_{i=1}^{4}B_{i}=1, we derive some new evolutionary structures of convex quadrilaterals and we give the definition of the degree of plasticity of convex quadrilaterals which could be extended to the degree of plasticity of convex polygons with respect to the topology of weighted Steiner minimal trees. Finally, the solution of the weighted Steiner tree problem for convex quadrilaterals gives a second property, which is the translation between the two Fermat-Torricelli points.

Author information

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

Anastasios Zachos

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

azachos@gmail.com

Keywords

  • Weighted Fermat-Torricelli problem
  • Steiner minimal tree
  • convex quadrilaterals

Mathematics Subject Classification

51E12, 52A10, 52A55, 51E10

References

  1. 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
  2. K. Bopp: Über das kürzeste Verbindungssystem zwischen vier Punkten, Dissertation, Göttingen (1879).Google Scholar
  3. D. Cieslik: Shortest Connectivity. An Introduction with Applications in Phylogeny, Springer, New York (2005).Google Scholar
  4. R. Courant, H. Robbins: What is Mathematics? Oxford University Press, New York (1941).Google Scholar
  5. E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. (1968) 1–29.https://doi.org/10.1137/0116001Google Scholar
  6. 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
  7. A. Zachos: A weighted Steiner minimal tree for convex quadrilaterals on the two-dimensional K-plane, J. Convex Analysis 18(1) (2011) 139–152.https://doi.org/10.68381/jca18007Google Scholar

Suggested citation

A. Zachos, G. Zouzoulas. “An Evolutionary Structure of Convex Quadrilaterals. Part II.” Journal of Convex Analysis 20 (2013), No. 2, 483–493. https://doi.org/10.68381/jca20029

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.