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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Centers and nearest points of sets


Authors: P. Szeptycki and F. S. Van Vleck
Journal: Proc. Amer. Math. Soc. 85 (1982), 27-31
MSC: Primary 46B99; Secondary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1982-0647891-6
MathSciNet review: 647891
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Abstract: For a Banach space $ X$ and a subset $ A$ of $ X$, $ {c_A}$ denotes the Čebyšev center of $ A$ and $ {P_A}x$ denotes the nearest point in $ A$ to the point $ x$ in $ X$. The space of all subsets of $ X$ is furnished with the Hausdorff metric. The modulus of continuity of the function $ A \to {c_A}$ is computed in the case when $ X$ is a Hilbert space and the sets $ A$ are compact; the same is done for the function $ A \to {P_A}x$, for fixed $ x$, in the case when $ X$ is uniformly convex and the sets $ A$ are convex and closed.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0647891-6
Keywords: Čebyšev center, nearest point, modulus of continuity
Article copyright: © Copyright 1982 American Mathematical Society