Chebyshev centers in normed spaces

Abstract

Let E be a normed linear space, F a bounded subset, Y a closed subset of E. A nonnegative real number r,,(F) is called the relative Chebyshev radius of F with respect to Y if r,,(F) is the inlimum of all numbers r > 0 for which there exists a y E Y such that F is contained in the closed ball B(v, r) with center y and radius r. Any point y E Y for which F c B(y, r,,(F)) is called a relative Chebyshev center of F with respect to Y. We denote the set of all relative Chebyshev centers of F with respect to Y by zy(F). In this paper we investigate several questions concerning characterization and existence of relative Chebyshev centers, and the continuity of the Chebyshev center map. In Section 1 we give a formula for the relative Chebyshev radius of a bounded set F with respect to Y in terms of the relative radius of F with respect to hyperplanes from the annihilator of Y. For F totally bounded this formula was obtained in [8]. Let F be a bounded set which is contained in the closed ball B(y, r), where r = r(y, F) 3 sup { I] x y )I ; x E F}. In Section 2 we are looking for necessary and sufficient conditions for B(y, r) to be the Chebyshev ball of F. For Hilbert space, a characterization was given in [5]. However, the necessity part of this characterization requires a property valid only for F compact. We give necessary and sufficient conditions for both the compact and noncompact case and deduce several corollaries. In Section 3 we show that every infinite dimensional normed space E has an equivalent norm such that c,(E) does not admit relative centers for all pairs of points in l,(E). In Section 4 we investigate the Lipschitz constants of the Chebyshev center map restricted to certain families of “admissible” pairs of sets, as introduced in [5].