
Journal of Convex Analysis 20 (2013), No. 2, 483493 Copyright Heldermann Verlag 2013 An Evolutionary Structure of Convex Quadrilaterals. Part II Anastasios Zachos Dept. of Mathematics, University of Patras, 26500 Rion, Greece azachos@gmail.com Gerasimos Zouzoulas G. Zouzoulas Ltd, Meintani 25, 11741 Athens, Greece pantarei@zouzoulas.gr [Abstractpdf] [For Part I see the authors, J. Convex Analysis 15 (2008) 411426.] We solve explicitly the generalized Gauss problem for convex quadrilaterals in the two dimensional Euclidean Space. By introducing the variable $$ c=c_{G}+\frac{B_{1}B_{4}+B_{2}B_{3}}{2}, \quad\text{where}\quad c_{G}=\frac{1}{2} $$ is the Gauss constant and $B_{i}$ are positive real variables, such that $\sum_{i=1}^{4}B_{i}=1,$ 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 FermatTorricelli points. Keywords: Weighted FermatTorricelli problem, Steiner minimal tree, convex quadrilaterals. MSC: 51E12, 52A10, 52A55, 51E10 [ Fulltextpdf (144 KB)] for subscribers only. 