2014/11/06 by Johnson, Marianne, Kambites, Mark · 1 citation
#14T05 #16Y60 #52B11 #Algebraic Geometry (math.AG) #FOS: Mathematics #Metric Geometry (math.MG) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1411.1630
We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of ℝn. We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovic and show that row spaces of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig and Kulas.