Chebyshev centers, \(\epsilon\)-Chebyshev centers and the Hausdorff metric.

*(English)*Zbl 0675.41042A 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.

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 |

##### Keywords:

p-uniformly convex; convexity modulu; Chebyshev radii; Chebyshev centers; Hausdorff distance; Lipschitz stability of \(\epsilon\)-Chebyshev centers##### References:

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