A metric on max-min algebra

Using the characterization of the segments in the max-min semi-module B-n, provided by Nitica and Singer in Contributions to max-min convex geometry. I: Segments. Linear Algebra and its Applications 428, (2008), 1439-1459, we find a class of metrics on the B-n One of them is given by the Euclidean length of the max-min segment connecting two points. The max-min segments are complicated and consist of several Euclidean segments pointing in a finite number of fixed directions. The number of directions increases with the dimension of the semimodule. Each metric in our class is associated with a weighting function, for which we give some characterization. None of these metrics is a quasiconvex metric. Nevertheless, a somehow weaker condition always holds.