Max-plus convex sets and max-plus semispaces. II

Abstract

We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I. Optimization, 56, 171-205. We show that the theories of max-plus convexity in R n max and B-convexity in R n þ are equivalent, and we deduce some consequences. We show that max-plus convexity in R n is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unless is a total order on A, the results of loc. cit. on semispaces cannot be generalized in a natural way to the framework of A n ¼ ðA n , ; Þ, where A :¼ M [ fÀ1g, with M ¼ ðM, ; Þ being a lattice ordered group and À1 a ''least element'' adjoined to M.

Characterization of tropical hemispaces by(P,R)-decompositions

“…If 0 < α ≤ β then by Lemma 3.15 we have Remark 3.18. The notation for sectors and semispaces is reversed as compared to the notation in Nitica and Singer [23,24,25].…”

“…Section 2 is occupied with preliminaries on convex sets in T n , and introduces the concept of (P, R)-decomposition. In Section 3 we study semispaces in T n , in order to give, exploiting homogenization, a simpler proof of their characterization than the one given in [23,24]. Hemispaces appear here as unions of (in general, infinitely many) complements of semispaces, i.e., the closed sectors of [16].…”

“…Next, an important feature of tropical convexity (as opposed to usual convexity) is the existence of a finite number of types of semispaces, i.e., maximal convex sets in T n not containing a given point. These sets were described in detail by Nitica and Singer [23,24,25], who showed that they are precisely the complements of closed sectors. Let us mention that the multiorder principle of tropical convexity [23,24,26,21] can be formulated in terms of complements of semispaces.It follows from abstract convexity that any hemispace is the union of all the complements of semispaces which it contains.…”

The geometric structure of max-plus hemispaces

“…We observe that the notion of face that we use can be extended to include the notion of (max-plus) sector, related to the complement of a semispace, that appeared before in the max-plus literature [4,7,11,12,13,8]. For a fixed supporting hyperplane, the conical decomposition of the boundary extends to a conical decomposition of the whole space R n max .…”

“…Our work can be viewed as a complement to the recent results of Katz-Nitica-Sergeev, who described generating sets for maxplus hemispaces, and the results of Briec-Horvath, who proved that closed/open max-plus hemispaces are max-plus closed/open halfspaces.= {max(α + x, β + y)|α, β ∈ R max , max(α, β) = 0}.(1)We recall several classes of convex sets that will be used in the sequel.Remark 1.4. Semispaces are introduced in [11,12]. It is shown in [11] that semispaces form an intersectional basis for the collection of convex sets.Definition 1.5.…”