Chebyshev sets and ball operators (Q2922465)

Let \(K\) be a bounded set in a finite-dimensional real normed linear space. The union of centers of all minimal enclosing balls of \(K\) is called the Chebyshev set of \(K\), and its elements are called Chebyshev centers of \(K\). In this paper, the authors derive some results on Chebyshev sets and Chebyshev centers in real Banach spaces of finite dimension. These results show 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 \(K\). For a special class of sets viz. centrable sets, the authors derive a necessary and sufficient condition that the Chebyshev set is a singleton. A complete geometric description of the ball hull of a finite set in the plane is also given, which is useful for algorithmical constructions of the ball hull of such sets.