Chebyshev centers and uniform convexity

If E is a uniformly convex Banach space and T is any topological space, then in the space X → C(T, E) of E-valued bounded continuous functions on E, every bounded set has a Chevyshev center. Moreover, the set function A →Z(A), corresponding to A the set of its Chebyshev centers, is uniformly continuous on bounded subsets of the space (X) of bounded subsets of X with the Hausdorff metric. This is contrasted with the fact that a normed space X in which Z(A) is a singleton for every bounded A is uniformly convex iff A→Z(A) is uniformly continuous on bounded subsets of B(X).