Characterization of tropical hemispaces by (P,R)-decompositions

TL;DR

This paper introduces a novel (P,R)-decomposition for tropical hemispaces, providing a new representation of tropically convex sets and characterizing hemispaces via associated matrices satisfying rank-one conditions.

Contribution

It presents the first (P,R)-decomposition framework for tropical hemispaces, extending classical convex set representations and linking them to matrix conditions in tropical geometry.

Findings

Tropical hemispaces are characterized by (P,R)-decompositions.

Associated matrices satisfy an extended rank-one condition.

The approach uses homogenization and the relation between hemispaces and semispaces.

Abstract

We consider tropical hemispaces, defined as tropically convex sets whose complements are also tropically convex, and tropical semispaces, defined as maximal tropically convex sets not containing a given point. We introduce the concept of $(P,R)$-decomposition. This yields (to our knowledge) a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays. We characterize tropical hemispaces as tropically convex sets that admit a (P,R)-decomposition of certain kind. In this characterization, with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield, satisfying an extended rank-one condition. Our proof techniques are based on homogenization (lifting a convex set to a cone), and the relation between tropical hemispaces and semispaces.