Abstract
Many problems in continuous location theory, reduce to finding a best location, in the sense that a facility must be located at a point minimizing the sum of distances to the points of a given finite set (median) or the largest distances to all points (center). The setting is often assumed to be a Banach space.
To have a better understanding concerning the structure of location problems, it may occur also in rather simple cases.
In this paper we indicate two simple examples of four-point sets such that one of the two problems indicated has a solution, while the other one has no solution. Also, we list papers containing examples previously given, dealing with this lack of optimal solutions.
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References
Amir D., Mach J. and Saatkamp K. (1982). Existence of Chebyshev centers, best n-nets and best compact approximants.Transactions of the American Mathematical Society 271, 513–524.
Baronti M., Casini E. and Papini P.L. (1993). Equilateral sets and their central points.Rendiconti di Matematica e delle Sue Applicazioni 13, 133–148.
Benítez C., Fernández M. and Soriano M.L. (2002). Location of the Fermat-Torricelli medians of three points.Transactions of the American Mathematical Society 354, 5027–5038.
Garkavi A.L. (1964). The best possible net and the best possible crosss-section of a set in a normed space.Transactions of the American Mathematical Society 39, 111–132.
Veslý L. (1993). A characterization of reflexivity in the terms of the existence of generalized centers.Extracta Mathematicae 8, 125–131.
Veselý L. (1997). Generalized centers of finite sets in Banach spaces.Acta Mathematica Universitatis Comenianae 66, 83–115.
Veselý L. (2002). A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers.Archiv der Mathematik 79, 499–506.
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Papini, P.L. Two new examples of sets without medians and centers. Top 13, 315–320 (2005). https://doi.org/10.1007/BF02579057
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DOI: https://doi.org/10.1007/BF02579057