Abstract
In this paper, we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. We obtain a characterization of two-dimensional real strictly convex spaces as those ones where a Chebyshev center cannot contribute to the set of farthest points of a subset. In dimension greater than two, every non-Hilbert smooth space contains a subset whose Chebyshev center is a farthest point. We explore the scenario in uniformly convex Banach spaces and further study the roles played by centerability and Mcompactness in the scheme of things to obtain a step by step characterization of strictly convex Banach spaces.
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The first author lovingly acknowledges the constant support and encouragement coming from his beloved younger brother Mr. Debdoot Sain.
The research of the second author is done in frames of Ukrainian Ministry of Science and Education Research Program 0115U000481; partially it was done during his stay in Murcia (Spain) under the support of MINECO/FEDER project MTM2014-57838-C2-1-P and Fundación Séneca, Región de Murcia grant 19368/PI/14, and partially during his visit to the University of Granada which was supported by MINECO/FEDER project MTM2015-65020-P.
The fourth author would like to thank DST, Govt. of India, for the financial support in the form of doctoral fellowship.
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Sain, D., Kadets, V., Paul, K. et al. Chebyshev Centers that are Not Farthest Points. Indian J Pure Appl Math 49, 189–204 (2018). https://doi.org/10.1007/s13226-018-0262-y
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DOI: https://doi.org/10.1007/s13226-018-0262-y