The tropical analogue of polar cones

TL;DR

This paper develops a tropical analogue of the classical polar and bipolar theorems, providing new characterizations of tropical convex cones and their inequalities using separation theorems.

Contribution

It introduces a tropical polar concept, proves a bipolar theorem, and extends results to systems of linear equalities, advancing tropical convex analysis.

Findings

Established a tropical bipolar theorem.

Derived a new separation theorem for tropical convex cones.

Extended results to linear equality systems in tropical geometry.

Abstract

We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.