Some stability properties of restricted Chebyshev centers in Banach spaces

In this paper, we discuss the stability of (restricted) Chebyshev centers in few function spaces. For an extremally disconnected compact Hausdorff space $K$ and a finite dimensional Banach space $X$, we prove the existence of Chebyshev centers of closed bounded subsets of the space of $X$-valued continuous functions on $K$, $C(K,X)$ and that of compact subsets of $M$-ideals in $C(K,X)$. It is also proved that the existence of restricted Chebyshev centers is stable in the space of vector-valued bounded functions on an arbitrary set. Furthermore, the stability of continuity properties of the Chebyshev-center map of a Banach space $X$ in dependence on the continuity properties of the Chebyshev-center map of $C(K,X)$ is studied.