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Location Problems with Different Norms for Different Points

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Abstract

Given a finite set \(A=\{a_1,\dots,{a_n}\}\) in a linear space X, we consider two problems. The first problem consists of finding the points minimizing the maximum distance to the points in A; the second problem looks for the points that minimize the average distance to the points in A. In both cases, we assume that the distances at different points are defined as

$$ d(x,{a_i})=\|x-{a_i}\|_i,\quad for i=1,\dots,n, $$

with norms \(\|\cdot \|_i\) defined on X. The use of different norms to measure distances from different points allows us to extend some results that hold in the single-norm case, while some strange and rather unexpected facts arise in the general case.

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Authors and Affiliations

  1. Professor,Department of Mathematics, University of Bologna, Bologna, Italy

    P. L. Papini

  2. Professor, Faculty of Mathematics, University of Seville, Seville, Spain

    J. Puerto

Authors
  1. P. L. Papini
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  2. J. Puerto
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Additional information

The research of the first author was supported by the Italian National Group GNAMPA.

The research of the second author was supported by the Spanish Ministry of Science and Technology through Grant BFM2001-2378, MTM2004:0909

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Papini, P.L., Puerto, J. Location Problems with Different Norms for Different Points. J Optim Theory Appl 125, 673–695 (2005). https://doi.org/10.1007/s10957-005-2095-1

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  • DOI: https://doi.org/10.1007/s10957-005-2095-1

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