Journal of Convex Analysis 21 (2014), No. 3, 601--618
Copyright Heldermann Verlag 2014
Chebyshev Sets and Ball Operators
Pedro Martín
Dep. de Matemáticas, Universidad de Extremadura, 06006 Badajoz, Spain
pjimenez@unex.es
Horst Martini
Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany
horst.martini@mathematik.tu-chemnitz.de
Margarita Spirova
Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany
margarita.spirova@mathematik.tu-chemnitz.de
The Chebyshev set of a bounded set K in a normed space is the set of centers of all minimal enclosing balls of K. We use the concept of ball intersection and ball hull operators to derive new properties of Chebyshev sets in normed spaces. These results give a better picture on how Chebyshev sets, ball intersections, ball hulls, and completions of bounded sets are related to each other. It is shown that the Chebyshev set of a bounded set K always contains the Chebyshev set of some completion of K. Moreover, for a special class of sets we obtain a necessary and sufficient condition that the Chebyshev set of the respective set is a singleton. We obtain new results on critical sets of Chebyshev centers, and for that purpose, surprisingly, notions from the combinatorial geometry of convex bodies play an essential role. Also we give a complete geometric description of the ball hull of a finite planar set. This can be taken as starting point for algorithmical constructions of the ball hull of such sets.
Keywords: Ball hull, ball intersection, Banach space, Chebyshev center, Chebyshev set, complete set, constant width, Jung's constant, Minkowski geometry, normed space, spherical intersection property.
MSC: 41A50, 41A61, 46B20, 52A21
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