Chebyshev centers and uniform convexity
Dan Amir · Pacific Journal of Mathematics · 1978
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 Let (X, d) be a metric space.Denoto by &(X) the space of nonempty bounded subsets of X and let h be the Hausdorff semimetric onFor x e X, r ^ 0, let B(x, r) = {y e X; d(x, y) ^ r} be the closed r-ball around x.For Ae&(X)and xeX denote r(x, A) = inf {r ^ 0; 7. R. B. Holmes, A course in optimization and best approximation, Springer