Ordered median functions and symmetries

Jerzy Grzybowski Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, PL-61-614 Poznań, Poland

Stefan Nickel Institute of Operations Research, University of Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany

Diethard Pallaschke Institute of Operations Research, University of Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, GermanyCorrespondence[email protected]

Ryszard Urbański Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, PL-61-614 Poznań, Poland

Abstract

In this article, we study symmetry properties of ordered median functions (S. Nickel and J. Puerto, Location Theory – A Unified Approach, Springer-Verlag, Berlin, Heidelberg, 2005). In particular, we prove that every ordered median function is a DCH-function (V.F. Demyanov and A.M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986; D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets – Fractional Arithmetic with Convex Sets, Mathematics and its Applications, Vol. 548, Kluwer Academic Publishers, Dordrecht, 2002), i.e. can be represented as a difference of two sublinear functions. Moreover, we give a necessary and sufficient condition for an ordered median function to be convex.

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Acknowledgements

The authors thank Jörg Kalcsics, Institute of Operations Research, University of Karlsruhe, for his very careful correction of the first version and an anonymous referee for his valuable remarks and his short and elegant proof of the necessary condition of Theorem 3.6.

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