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 ℬ(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).
