A Weighted Steiner Minimal Tree for Convex Quadrilaterals on the Two-Dimensional K-Plane

Anastasios Zachos

doi:10.68381/jca18007

Abstract

We provide a method to find a weighted Steiner minimal tree for convex quadrilaterals on a two-dimensional hemisphere of radius

\frac{1}{\sqrt{K}}

, for

K>0

and the two dimensional hyperbolic plane of constant Gaussian Curvature K, for

K<0

by introducing a method of cyclical differentiation of the objective function with respect to four variable angles. By applying this method, we find a generalized solution to a problem posed by C.F. Gauss in the spirit of weighted Steiner trees.

Author information

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

Anastasios Zachos

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

azachos@gmail.com

Keywords

Steiner minimal tree
generalized convex quadrilaterals

Mathematics Subject Classification

51E12, 52A10, 52A55, 51E10

References

I. D. Berg, I. G. Nikolaev: On an extremal property of quadrilaterals in an Aleksandrov space of curvature ≤ K, in: The Interaction of Analysis and Geometry (Novosibirsk, 2004), V. I. Burenkov et al. (ed.), Contemp. Math. 424, AMS, Providence (2007) 1–15.https://doi.org/10.1090/conm/424/08092 Google Scholar
V. Boltyanski, H. Martini, V. Soltan: Geometric Methods and Optimization Problems, Kluwer, Dordrecht (1999).https://doi.org/10.1007/978-1-4615-5319-9 Google Scholar
K. Bopp: Über das kürzeste Verbindungssystem zwischen vier Punkten, Dissertation, Universität Göttingen, Göttingen (1879).Google Scholar
D. Cieslik: Shortest Connectivity. An Introduction with Applications in Phylogeny, Springer, New York (2005).Google Scholar
R. Courant, H. Robbins: What is Mathematics?, Oxford University Press, New York (1951).Google Scholar
E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29.https://doi.org/10.1137/0116001 Google Scholar
A. N. Zachos, G. Zouzoulas: The weighted Fermat-Torricelli problem and an "inverse" problem, J. Convex. Analysis 15(1) (2008) 55–62.https://doi.org/10.68381/jca15004 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.

[1] I. D. Berg, I. G. Nikolaev: On an extremal property of quadrilaterals in an Aleksandrov space of curvature ≤ K, in: The Interaction of Analysis and Geometry (Novosibirsk, 2004), V. I. Burenkov et al. (ed.), Contemp. Math. 424, AMS, Providence (2007) 1–15.https://doi.org/10.1090/conm/424/08092 Google Scholar

[2] V. Boltyanski, H. Martini, V. Soltan: Geometric Methods and Optimization Problems, Kluwer, Dordrecht (1999).https://doi.org/10.1007/978-1-4615-5319-9 Google Scholar

[3] K. Bopp: Über das kürzeste Verbindungssystem zwischen vier Punkten, Dissertation, Universität Göttingen, Göttingen (1879).Google Scholar

[4] D. Cieslik: Shortest Connectivity. An Introduction with Applications in Phylogeny, Springer, New York (2005).Google Scholar

[5] R. Courant, H. Robbins: What is Mathematics?, Oxford University Press, New York (1951).Google Scholar

[6] E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29.https://doi.org/10.1137/0116001 Google Scholar

[7] A. N. Zachos, G. Zouzoulas: The weighted Fermat-Torricelli problem and an "inverse" problem, J. Convex. Analysis 15(1) (2008) 55–62.https://doi.org/10.68381/jca15004 Google Scholar