Tropical Halfspaces

Summary

AS a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R, min, +). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the understanding is a newly defined sign of the tropical determinant, which shares remarkably many properties with the ordinary sign of the determinant of a matrix. The methods are used to obtain an optimal tropical convex hull algorithm in two dimensions.

1. Introduction

The set R of real numbers carries the structure of a semiring if equipped with the tropical addition min﹛ and the tropical multiplication where + is the ordinary addition. We call the triplet the tropical semiring. It is an equally simple and important fact that the operations are continuous with respect to the standard topology of R. So the tropical semiring is, in fact, a topological semiring. Considering the tropical scalar multiplication (and componentwise tropical addition) turns the set [R^d+1 into a semimodule. The study of the linear algebra of the tropical semiring and, more generally, of idempotent semirings, has a long tradition. Applications to combinatorial optimization, discrete event systems, functional analysis etc. abound. For an introduction see [Baccelli et al. 1992]. A recent contribution in the same vein, with many more references, is [Cohen et al. 2004].

Convexity in the tropical world (and even in a more general setting) was first studied by Zimmermann [1977]. Following the approach of Develin and Sturmfels [2004] here we stress the point of view of discrete geometry.