Signed tropical convexity

TL;DR

This paper introduces a novel concept of tropical convexity for signed tropical numbers, providing multiple characterizations, and extending classical convexity theorems to this new setting.

Contribution

It develops the theory of signed tropical convexity, including a new Farkas lemma, Fourier-Motzkin elimination, and a Minkowski-Weyl theorem for signed tropical polytopes.

Findings

New notion of tropical convexity for signed numbers

Equivalent descriptions involving balance relations and intersections

Minkowski-Weyl theorem for signed tropical polytopes

Abstract

We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.