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

The max-plus semifield R-max is the set R boolean OR {-infinity} with the addition a circle plus b := max(a, b) and the multiplication a circle plus b := a + b. A subset C of R-max(n) is said to be max-plus convex if the relations x, y epsilon C,alpha, beta epsilon R-max, alpha circle plus beta = 0 (the unit element of circle times) imply (alpha circle times x) circle plus (beta circle times gamma) epsilon C, where circle plus is understood componentwise and alpha circle times x:= (alpha circle times x(1),..., alpha circle times x(n)) for alpha epsilon R-max, x = (x(1),..., x(n)) epsilon R-max(n). 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 z epsilon R-max(n) is a maximal (with respect to inclusion) max-plus convex subset of R-max(n)\{z}. In contrast to the case or linear spaces, where there exist all infinity of semispaces at each point, we show that in R-max(n) 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 ill (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).