Max-plus convex sets and max-plus semispaces. I‡

Abstract

The max-plus semifield R max is the set with the addition a⊕ b:=max(a,b) and the multiplication a⊗ b:=a+b. A subset C of is said to be max-plus convex if the relations (the unit element of ⊗) imply (α⊗x)⊕(β⊗y)∈C, where ⊕ is understood componentwise and for In analogy with the definition of semispaces for usual linear spaces (see e.g. Hammer (Hammer, P.C., 1955, Maximal convex sets. Duke Mathematical Journal, 22, 103–106)), a max-plus semispace at a point is a maximal (with respect to inclusion) max-plus convex subset of In contrast to the case of linear spaces, where there exist an infinity of semispaces at each point, we show that in there exist at most n + 1 max-plus semispaces at each point, and exactly n + 1 at each point whose all coordinates are finite. We determine these max-plus semispaces and give some consequences for separation of max-plus convex sets from outside points. Some different separation theorems for closed max-plus convex sets were given in (Cohen, G., Gaubert, S., Quadrat, J.-P. and Singer, I., 2005, Max-plus convex sets and functions. Contemporary Mathematics, 377, 105–129, circulated previously as Preprint 1341/2003, ESI, Vienna, and arXiv:math.FA/0308166). ‡The authors presented this article at the 8th Symposium on Generalized Convexity/Monotonicity held in Varese, July 4–8, 2005.