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

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 image endowed with the operations circled plus=max,circle times operator=min in image A subset C of image is said to be max–min convex if the relations image (the neutral element of circle times operator) imply (αcircle times operatorx)circled plus(βcircle times operatory)set membership, variantC, where circled plus is understood componentwise and αcircle times operatorxcolon, equals(αcircle times operatorx1,…,αcircle times operatorxn) for image. 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 image is a maximal (with respect to inclusion) max–min convex subset of image. In contrast to the case of linear spaces, where there exist an infinity of semispaces at each point, we show that in image 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 image is a multi-order convexity.