Search

We prove that in finite dimensional spaces every ordered median function is the Minkowski dual of a reduced pair of polytopes. This implies a very general theorem on the representation of an ordered median function as a uniquely determined difference of two sublinear functions up to adding and subtracting one and the same arbitrary sublinear function. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article we give, for an arbitrary continuous function which is linear on the octants of [image omitted] an explicit representation in terms of a max-min expression. Moreover, we also consider its representation as the difference of two sublinear functions and determine minimal representations in the sense of Pallaschke and Urbanski [D. Pallaschke and R. Urbanski, Pairs of Compact Convex Sets - Fractional Arithmetic with Convex Sets, Vol. 548, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2002]. [ABSTRACT FROM AUTHOR]

In this paper, we determine the cardinality of various subsets of the family of all minimal pairs of compact convex subsets of n (Theorems 2.1 and 2.8). Furthermore, we express minimality properties of a pair of closed bounded convex sets in set-theoretical terms (Theorem 3.3). [ABSTRACT FROM AUTHOR]