Structure of Max-Plus Hemispaces

We recall from [1] that a line segment in Rmax, and consiquently Rmax has a similar case, has one of the three following forms: When X ≤ Y and x1 − y1 ≤ x2 − y2, then [X,Y ] = [X, (y1 + x2 − y2, x2)] ∪ [(y1 + x2 − y2, x2), Y ] (2) When X ≤ Y and x2 − y2 ≤ x1 − y1, then [X,Y ] = [X, (x1, y2 + x1 − y1)] ∪ [(x1, y2 + x1 − y1), Y ] (3) When X Y and Y X, then [X,Y ] = [X,max(X,Y )] ∪ [max(X,Y ), Y ] (4)