Contributions to max–min convex geometry. II: Semispaces and convex sets

In this article, continuing [V. Nitica, I. Singer, Contributions to max–min convex geometry. I: Segments, Linear Algebra Appl. (2007), doi:10.1016/j.laa.2007.09.032], we give some further contributions to the theory of “max–min geometry”. The max–min semifield is the set endowed with the operations ⊕=max,⊗=min in A subset C of is said to be max–min convex if the relations (the neutral element of ⊗) imply (α⊗x)⊕(β⊗y)∈C, where ⊕ is understood componentwise and α⊗x≔(α⊗x1,…,α⊗xn) for . In analogy with the definition of semispaces for usual linear spaces (see e.g. [P.C. Hammer, Maximal convex sets, Duke Math. J. 22 (1955) 103–106]), a max–min semispace at a point is a maximal (with respect to inclusion) max–min 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–min semispaces at each point and exactly n+1 at each point whose all coordinates are finite. We determine these max–min semispaces and give some consequences for separation of max–min convex sets from outside points. We show that max–min convexity restricted to the finite part Rn of is a multi-order convexity.