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Chebyshev centers, \(\epsilon\)-Chebyshev centers and the Hausdorff metric. (English) Zbl 0675.41042
A Banach space X is called p-uniformly convex \((1<p<\infty)\), if there exists a constant \(c>0\) such that \(\delta_ X(\epsilon)\geq c\). \(\epsilon^ p\), for all \(\epsilon >0\), where \(\delta_ X\) denotes the convexity modulu of the space X.
The main result of this paper is the following: A Banach space X is p- uniformly convex if and only if there exists a constant \(C>0\) such that \[ \| x_ A-x_ B\|^ p\leq C((r_ B+h(A,B))^ p-r^ p_ A), \] for all pairs A,B of bounded subsets of X. Here \(r_ A,r_ B\) and \(x_ A,x_ B\) denote the Chebyshev radii respectively Chebyshev centers (with respect to X) of the sets A,B and h(A,B) the Hausdorff distance between A,B. This solves affirmatively a question raised by P. Szeptycki and F. S. van Vleck [Proc. Am. Math. Soc. 85, 27-31 (1982; Zbl 0511.41029)], who proved a similar estimation in the case of a Hilbert space X and compact subsets A,B of X. the Lipschitz stability of \(\epsilon\)-Chebyshev centers is also studied.
Reviewer: S.Cobzaş

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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