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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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References:
[1] D.AMIR Chebyshev centers and uniform convexity Pacific J. Math. 77, 1-6 (1978)
[2] ?. Approximation by certain subspaces in the Banach space of continuous vector valued functions. J. Approx. Th. 27, 254-270 (1979) · Zbl 0426.41027
[3] ? A note on ?Approximation of bounded sets? J. Approx. Th. 44, 92-93 (1985) · Zbl 0603.41025
[4] H.ATTOUCH & J.B.-WETS Lipschitzian stability of ?-approximate solutions in convex optimization. (preprint)
[5] B.BEAUZAMY Introduction to Banach spaces and their geometry. North-Holland Math. Stud.(68) second edition · Zbl 0491.46014
[6] J.H.FREILICH & H.W.MCLAUGHLIN Approximation of bounded sets. J. Approx. Th. 34, 146-158 (1982) · Zbl 0504.41013
[7] O.HANNER On the uniform convexity of Lp and lp Ark. Math. 3, 239-244 (1956) · Zbl 0071.32801
[8] LI CHONG On a problem on Chebyshev centers (1987) submitted
[9] P.SZEPTYCKI & F.S.VAN VLECK Centers and nearest points of sets. Proc. A.M.S. 85, 27-31 (1982) · Zbl 0511.41029
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