Chebyshev centers in normed spaces (Q797765)

Let E be a normed linear space, F a bounded subset, Y a closed subset of E. A nonnegative real number \(r_ Y(F)\) is called the relative Chebyshev radius of F with respect to Y if \(r_ Y(F)\) is the infimum of all numbers \(r>0\) for which there exists a \(y\in Y\) such that F is contained in the closed ball B(y,r) with center y and radius r. Any point \(y\in Y\) for which \(F\subset B(y,r_ Y(F))\) 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 \(z_ Y(F).\) The paper investigates several questions concerning characterization and existence of relative Chebyshev centers, and the continuity of the Chebyshev center map. Section 1 gives 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, extending a result of Franchetti and Cheney. Let F be a bounded set which is contained in the closed ball B(y,r), where \(r=r(y,F)\equiv\sup \{\| x-y\|\); \(x\in F\}\). Section 2 gives necessary and sufficient conditions for B(y,r) to be the Chebyshev ball of F, including an alternative proof of the Laurent-Tuan characterization of the Chebyshev center. In Section 3 it is shown that every infinite dimensional normed space E has an equivalent norm such that \(c_ 0(E)\) does not admit relative centers for all pairs of points in \(\ell_{\infty}(E)\). Section 4 investigates the Lipschitz constants of the Chebyshev center map restricted to certain families of ''admissible'' pairs of sets, as introduced by Borwein and Keener. Sections 5 and 6 discuss the upper semicontinuity of the Chebyshev center map and the proximinality of isomorphic images of proximinal subspaces.