Vol. 77, No. 1, 1978

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Chebyshev centers and uniform convexity

Dan Amir

Vol. 77 (1978), No. 1, 1–6

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 (X).

Mathematical Subject Classification 2000
Primary: 46B20
Secondary: 46E40
Received: 12 July 1977
Published: 1 July 1978
Dan Amir