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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

##### MSC:
 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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