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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

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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