Abstract :
A max-plus hyperplane (briefly, a hyperplane) is the set of all points image satisfying an equation of the form a1x1circled pluscdots, three dots, centeredcircled plusanxncircled plusan+1=b1x1circled pluscdots, three dots, centeredcircled plusbnxncircled plusbn+1, that is, max(a1+x1,…,an+xn,an+1)=max(b1+x1,…,bn+xn,bn+1), with ai,biset membership, variantRmax(i=1,…,n+1), where each side contains at least one term, and where ai≠bi for at least one index i. We show that the complements of (max-plus) semispaces at finite points zset membership, variantRn are “building blocks” for the hyperplanes in image (recall that a semispace at z is a maximal – with respect to inclusion – max-plus convex subset of image. Namely, observing that, up to a permutation of indices, we may write the equation of any hyperplane H in one of the following two forms:imagewhere 0less-than-or-equals, slantpless-than-or-equals, slantqless-than-or-equals, slantmless-than-or-equals, slantn and all ai(i=1,…,m,n+1) are finite, or,imagewhere 0less-than-or-equals, slantpless-than-or-equals, slantqless-than-or-equals, slantmless-than-or-equals, slantn, and all ai(i=1,…,m) are finite (and an+1 is either finite or -∞), we give a formula that expresses a nondegenerate strictly affine hyperplane (i.e., with m=n and an+1>-∞) as a union of complements of semispaces at a point zset membership, variantRn, called the “center” of H, with the boundary of a union of complements of other semispaces at z. Using this formula, we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior. We give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center, and we characterize the cases in which the boundary bd H of a nondegenerate strictly affine hyperplane H is also a hyperplane. Next, we give the relations between nondegenerate strictly affine hyperplanes H, their centers z, and their coefficients ai. In the converse direction we show that any union of complements of semispaces at a point zset membership, variantRn with the boundary of any union of complements of some other semispaces at that point z, is a nondegenerate strictly affine hyperplane. We obtain a formula for the total number of strictly affine hyperplanes. We give complete lists of all strictly affine hyperplanes for the cases n=1 and n=2. We show that each linear hyperplane H in image (i.e., with an+1=-∞) can be decomposed as the union of four parts, where each part is easy to describe in terms of complements of semispaces, some of them in a lower dimensional space.
