Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 18 (2011), No. 1, 139--152Copyright 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 azachos@gmail.com [Abstract-pdf] 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 [ Fulltext-pdf  (129  KB)] for subscribers only.