In this work we study a relative Chebyshev center of K with respect to Y, where K is a closed bounded convex subset of a Hilbert space X, and Y is a closed convex subset of X. Some results of Amir and Mach [J. Approx. Theory 40, (1984), 364 374] are extended.
1997 Academic Press
We use the following notation and definitions: let X be a normed linear space, X* the dual of X, and F(X ) the set of all closed non-empty subsets of X; B(x, r)=[z # X: &z&x& r], S(x, r)=[z # X: &z&x&=r], S=S(0, 1), and
is the infimum of all numbers r>0 for which there exists y # Y such that K is contained in the ball B( y, r). Any point y # Y for which K/B( y, R Y (K )) is called a relative Chebyshev center of K with respect to Y. We denote the set of all relative Chebyshev centers of K with respect to Y by
, and P Y is the metric projection onto Y.
In this work we prove some assertions concerning characterization of relative Chebyshev centers. In Section 1 the main result of the paper is established that in Hilbert space X for a convex Y # F(X ) and a convex bounded K # F(Y) the relation Z Y (K ) # P Y (K) holds. This extends Corollary 2.9 of Amir and Mach [1], who assume that K is compact and convex. In Section 2 we give necessary and sufficient conditions for y to be the relative Chebyshev center of a convex bounded K # F(X ) with respect to a convex Y # F(X ), provided that X is Hilbert. When Y is a subspace, these results are equivalent to those of [1].