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This paper summarizes results on some topics in the max-plus convex geometry,mainly concerning the role of multiorder, Kleene stars and cyclic projectors,and relates them to some topics in max algebra. The multiorder principle leadsto max-plus analogues of some statements in the finite-dimensional convexgeometry and is related to the set covering conditions in max algebra. Kleenestars are fundamental for max algebra, as they accumulate the weights ofoptimal paths and describe the eigenspace of a matrix. On the other hand, theapproach of tropical convexity decomposes a finitely generated semimodule intoa number of convex regions, and these regions are column spans of uniquelydefined Kleene stars. Another recent geometric result, that several semimoduleswith zero intersection can be separated from each other by max-plus halfspaces,leads to investigation of specific nonlinear operators called cyclicprojectors. These nonlinear operators can be used to find a solution tohomogeneous multi-sided systems of max-linear equations. The results arepresented in the setting of max cones, i.e., semimodules over the max-timessemiring.

Multiorder, Kleene stars and cyclic projectors in the geometry of max cones