Chebyshev centers and uniform convexity

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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).

Original languageEnglish
Pages (from-to)1-6
Number of pages6
JournalPacific Journal of Mathematics
Issue number1
StatePublished - 1978


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