Journal of Convex Analysis 15 (2008), No. 2, 411--426
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, 117-41 Athens, Greece
pantarei@zouzoulas.gr
We solve the problem of the evolution of convex quadrilaterals by applying the inverse weighted Fermat-Torricelli problem, the invariance property of the weighted Fermat-Torricelli point in the plane R2, two-dimensional sphere S2 and the two-dimensional hyperboloid H2. This means that the property of plasticity is inherited by some evolutionary convex quadrilaterals. An important application is the connection of the Fermat-Torricelli point with the fundamental equation of P. de Fermat.
Keywords: Fermat-Torricelli problem, inverse Fermat-Torricelli problem, generalized convex quadrilaterals.
MSC: 51E12, 52A10, 52A55, 51E10
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