
Journal of Convex Analysis 20 (2013), No. 4, 10431073 Copyright Heldermann Verlag 2013 An Evolutionary Structure of Convex Pentagons on a C^{2} Complete Surface and a Creation Principle of some Weighted Dendrites of Order Three Anastasios N. Zachos University of Patras, Dept. of Mathematics, 26500 Rion, Greece azachos@gmail.com We reveal some new evolutionary structures of convex pentagons on a C^{2} surface in the three dimensional Euclidean Space which is hidden on the notion of plasticity of weighted convex pentagons and quadrilaterals. We prove a creation principle for a weighted dendrite of order 3 on a C^{2} surface which is derived by decreasing the degree of plasticity of weighted convex pentagons from 2 to 1. The creation principle of weighted dendrites of order 3 characterize the process of creation of various leaf tree types on surfaces. Keywords: Weighted FermatTorricelli problem, Steiner minimal tree, convex pentagons, surface, dendrite. MSC: 51E12, 52A10, 52A55, 51E10 [ Fulltextpdf (261 KB)] for subscribers only. 