
Journal of Convex Analysis 15 (2008), No. 2, 411426 Copyright Heldermann Verlag 2008 An Evolutionary Structure of Convex Quadrilaterals Anastasios N. Zachos Dept. of Mathematics, University of Patras, 26500 Rion, Greece zaxos@master.math.upatras.gr Gerasimos Zouzoulas Meintani 25, 11741 Athens, Greece pantarei@zouzoulas.gr We solve the problem of the evolution of convex quadrilaterals by applying the inverse weighted FermatTorricelli problem, the invariance property of the weighted FermatTorricelli point in the plane R^{2}, twodimensional sphere S^{2} and the twodimensional hyperboloid H^{2}. This means that the property of plasticity is inherited by some evolutionary convex quadrilaterals. An important application is the connection of the FermatTorricelli point with the fundamental equation of P. de Fermat. Keywords: FermatTorricelli problem, inverse FermatTorricelli problem, generalized convex quadrilaterals. MSC: 51E12, 52A10, 52A55, 51E10 [ Fulltextpdf (1973 KB)] for subscribers only. 