Journal of Convex Analysis 20 (2013), No. 4, 1043--1073
Copyright Heldermann Verlag 2013
An Evolutionary Structure of Convex Pentagons on a C2 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 C2 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 C2 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 Fermat-Torricelli problem, Steiner minimal tree, convex pentagons, surface, dendrite.
MSC: 51E12, 52A10, 52A55, 51E10
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