Hahn-Banach separation theorem for max-plus semimodules

We introduce max-plus analogues of basic Euclidian geometry notions: scalar product is replaced by a scalar division, and the associated distance is essentially Hilbert's projective distance. We introduce an orthogonal projection and prove a Hahn-Banach type theorem: a point can be separated from a semimodule by a hyperplane orthogonal to the direction of projection. We use these results to separate max-plus convex sets, and illustrate this new geometry by two-dimensional examples.