The Geometric Structure of Max-Plus Hemispaces

Abstract

Given a set S endowed with a convexity structure, a hemispace is a convex subset of S which has convex complement. We recall that R^n_{max} is a semimodule over the max-plus semifield. A convexity structure of current interest is provided by R^n_{max} naturally endowed with the max-plus (or tropical) convexity. In this paper we provide a geometric description of a max-plus hemispace. We show that a max-plus hemispace has a conical decomposition as a finite union of disjoint max-plus cones. These cones can be interpreted as faces of several max-plus hyperplanes. Briec-Horvath proved that the closure of a max-plus hemispace is bounded by a max-plus hyperplane. Given a hyperplane, we give a simple condition for the assignment of the faces between a pair of complementary max-plus hemispaces. Our result allows for counting and enumeration of the associated max-plus hemispaces. We recall that an n-dimensional max-plus hyperplane is called strictly affine and nondegenerate if it has a linear equation that contains all variables x_1,x_2,...,x_n and a free term. We prove that the number of max-plus hemispaces in R^n_max, supported by strictly affine nondegenerate hyperplanes centered in the origin, is twice the n-th ordered Bell number. Our work can be viewed as a complement to the recent results of Katz-Nitica-Sergeev, who described generating sets for max-plus hemispaces, and the results of Briec-Horvath, who proved that closed/open max-plus hemispaces are max-plus closed/open halfspaces.