On max–min representations of ordered median functions

Abstract

An ordered median function is a continuous piecewise linear function. It is well known, that in finite dimensional spaces every continuous piecewise linear function admits a max–min representation in terms of its linear functions. We give an explicit representation of an ordered median function in max–min form using a purely combinatorial approach.

Keywords:

• ordered median functions
• max–min-representation
• convex analysis

AMS Subject Classifications:

• 52A07
• 26B40
• 90B85

Acknowledgments

The authors thank an anonymous referee for his valuable remarks on the first version of the paper and his essential advise for a generalization of Proposition 1.2.