On hyperplanes and semispaces in max-min convex geometry

The concept of separation by hyperplanes is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to classically separate by hyperplanes in max-min convex geometry

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

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 leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by max-plus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multi-sided systems of max-linear equations. The results are presented in the setting of max cones, i.e., semimodules over the max-times semiring.Comment: 26 pages, a minor revisio

On the dimension of max-min convex sets

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvability of linear systems and trapezoidal property.Comment: 19 pages, v2: many corrections in the proof

An interval version of separation by semispaces in max-min convexity

We study separation of a closed box from a max-min convex set by max-min semispace. This can be regarded as an interval extension of known separation results. We give a constructive proof of the separation in the case when the box and the max-min convex set satisfy certain condition, and we show that separation is never possible if this condition does not hold. We also study separation of max-min convex sets by boxes and by box and semispace

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur

The tropical analogue of polar cones

We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.Comment: 21 pages, 3 figures, example added, figures improved, notation change