On the dimension of max–min convex sets

Abstract

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 the trapezoidal property.

(K,L)-eigenvectors in max-min algebra

“…The procedure also raises the question about the number of points that generate the λ-eigenspace. From Example 3.2, it is clear that the number of such points can exceed the dimension, as the eigenspace in this example is generated by (.4, .2), (.5, 1), (1, .5) (e.g., consider it as a "max-min quadrangle" between these points and the zero vector, and use the forms of max-min segments given in [24]), with none of these points being redundant. However, it is not known how quickly the minimal number of such generators can grow with matrix dimension.…”

“…Taking out the zero vector 0, we then obtain a generating set for the whole λ-eigenspace. See, e.g., Nitica and Sergeev [23,24] for more on max-min convexity.…”