Some remarks on relative Chebyshev centers (Q1359964)

Let \(X\) be a Hilbert space with \(F(X)\) as the set of all closed nonempty convex subsets of \(X\). Suppose that \(Y\), \(K\in F(X)\) and \(K\) is bounded. Then the author shows that the set of all relative Chebyshev centres of \(K\) with respect to \(Y\) is a subset of \(P_Y(K)\), where \(P_Y\) is the metric projection onto \(Y\). This extends an earlier result of \textit{D. Amir} and \textit{J. Mach} [J. Approximation Theory 40, 364-374 (1984; Zbl 0546.41029), Corollary 2.9], where \(K\) is assumed to be convex and compact. Necessary and sufficient conditions for relative Chebychev centres are also obtained.