
Journal of Convex Analysis 24 (2017), No. 1, 185198 Copyright Heldermann Verlag 2017 A Generalized FermatTorricelli Tree in the Three Dimensional Euclidean Space Anastasios N. Zachos Dept. of Mathematics, University of Patras, 26500 Rion, Greece azachos@gmail.com We obtain a generalized FermatTorricelli tree for a boundary pentagonal pyramid (closed hexahedron) and a boundary closed enniaedron in the three dimensional Euclidean Space by placing two mobile vertices with positive weights at the interior of their convex hull and by introducing a method of differentiation of the objective function with respect to two variable dihedral angles and four variable length of linear segments which connect each mobile vertex with two boundary vertices. The generalized FermatTorricelli tree with degree at most four with respect to a closed hexahedron or a closed enniaedron is an Euclidean minimal tree structure which may be considered as an intermediate class of a tree structure between a weighted FermatTorricelli (tree) structure and a weighted Steiner minimal tree structure with respect to a closed hexahedron or closed enniaedron. Keywords: Generalized FermatTorricelli tree, generalized FermatTorricelli constant, pyramids, closed polyhedra. MSC: 51E10, 52A15, 52B10 [ Fulltextpdf (134 KB)] for subscribers only. 