The structure of max-plus hyperplanes

Abstract

A max-plus hyperplane (briefly, a hyperplane) is the set of all points x = ( x 1 , … , x n ) ∈ R max n satisfying an equation of the form a 1 x 1 ⊕ ⋯ ⊕ a n x n ⊕ a n + 1 = b 1 x 1 ⊕ ⋯ ⊕ b n x n ⊕ b n + 1 , that is, max ( a 1 + x 1 , … , a n + x n , a n + 1 ) = max ( b 1 + x 1 , … , b n + x n , b n + 1 ) , with a i , b i ∈ R max ( i = 1 , … , n + 1 ) , where each side contains at least one term, and where a i ≠ b i for at least one index i. We show that the complements of (max-plus) semispaces at finite points z ∈ R n are “building blocks” for the hyperplanes in R max n (recall that a semispace at z is a maximal – with respect to inclusion – max-plus convex subset of R max n ⧹ { z } ) . 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: a 1 x 1 ⊕ ⋯ ⊕ a p x p ⊕ a p + 1 x p + 1 ⊕ ⋯ ⊕ a q x q = a 1 x 1 ⊕ ⋯ ⊕ a p x p ⊕ a q + 1 x q + 1 ⊕ ⋯ ⊕ a m x m ⊕ a n + 1 , where 0 ⩽ p ⩽ q ⩽ m ⩽ n and all a i ( i = 1 , … , m , n + 1 ) are finite, or, a 1 x 1 ⊕ ⋯ ⊕ a p x p ⊕ a p + 1 x p + 1 ⊕ ⋯ ⊕ a q x q ⊕ a n + 1 = a 1 x 1 ⊕ ⋯ ⊕ a p x p ⊕ a q + 1 x q + 1 ⊕ ⋯ ⊕ a m x m ⊕ a n + 1 , where 0 ⩽ p ⩽ q ⩽ m ⩽ n , and all a i ( i = 1 , … , m ) are finite (and a n + 1 is either finite or - ∞ ), we give a formula that expresses a nondegenerate strictly affine hyperplane (i.e., with m = n and a n + 1 > - ∞ ) as a union of complements of semispaces at a point z ∈ R n , 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 a i . In the converse direction we show that any union of complements of semispaces at a point z ∈ R n 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 R max n (i.e., with a n + 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.