
Journal of Convex Analysis 18 (2011), No. 3, 833853 Copyright Heldermann Verlag 2011 An Evolutionary Structure of Pyramids in the ThreeDimensional Euclidean Space Anastasios Zachos azachos@gmail.com Gerasimos Zouzoulas G. Zouzoulas Ltd, Meintani 25, 11741 Athens, Greece pantarei@zouzoulas.gr We solve the problem of evolution for some classes of pyramids in the three dimensional Euclidean space by applying the inverse weighted FermatTorricelli problem of 5 rays that meet at the weighted FermatTorricelli point A_{0} and the invariance property of the weighted FermatTorricelli point. The main result is the three dimensional property of plasticity which states that: If we decrease the weights that correspond to the first, third and fourth ray which passes from the apex of the pyramid, then the weights that correspond to the second and fifth ray increase. Finally, we introduce the notion of the generalized plasticity for weighted pyramids via a specific discretization of the five weights along the five given prescribed rays. Keywords: FermatTorricelli point, inverse FermatTorricelli problem, plasticity property, weighted pyramids. MSC: 51E10, 52A15, 52B10 [ Fulltextpdf (352 KB)] for subscribers only. 