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Title: On bisectors for convex distance functions
Authors: He, Chan
Martini, Horst
Wu, Senlin
Keywords: Birkhoff orthogonality;Bisector;Characterization of ellipse;Convex distance function;Euclidean norm;Gauge;Isosceles orthogonality;Roberts orthogonality;Voronoi diagram;Ortogonalidad de Birkhoff;Bisectriz;Caracterización de elipse;Función de distancia convexa;Norma euclidiana;Indicador;Ortogonalidad isósceles;Ortogonalidad de Roberts;Diagrama de Voronoi
Issue Date: 2013
Publisher: Universidad de Extremadura
Abstract: It is well known that the construction of Voronoi diagrams is based on the notion of bisector of two given points. Already in normed linear spaces, bisectors have a complicated structure and can, for many classes of norms, only be described with the help of topological methods. Even more general, we present results on bisectors for convex distance functions (gauges). Let C, with the origin o from its interior, be the compact, convex set inducing a convex distance function (gauge) in the plane, and let B( ̶ x, x) be the bisector of ̶ x and x, i.e., the set of points z whose distance (measured with the convex distance function induced by C) to ̶ x equals that to x. For example, we prove the following characterization of the Euclidean norm within the family of all convex distance functions: if the set L of points x in the boundary ∂C of C that create B( ̶ x, x) as a straight line has non-empty interior with respect to ∂C, then C is an ellipse centered at the origin. For the subcase of normed planes we give an easier approach, extending the result also to higher dimensions.
Appears in Collections:Extracta Mathematicae Vol. 28, nº 1 (2013)

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