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Journal of Convex Analysis 18 (2011), No. 1, 139--152
Copyright Heldermann Verlag 2011

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

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


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.

Keywords: Steiner minimal tree, generalized convex quadrilaterals.

MSC: 51E12, 52A10, 52A55, 51E10

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