Signed tropical halfspaces and convexity

TL;DR

This paper extends tropical convexity theory to signed numbers, introduces new convexity notions, and explores their properties and representations, linking them to oriented matroids and linear spaces over tropical hyperfields.

Contribution

It introduces the novel concept of TC-convexity, extends convexity beyond the positive orthant, and connects tropical convexity with oriented matroids and signed valuations.

Findings

Derived separation results for TO- and TC-convexity.

Characterized TC-hemispaces and their properties.

Linked TC-convexity to representations of oriented matroids.

Abstract

We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and V\'egh (ITCS 2020). We study two notions of convexity for signed tropical numbers called 'TO-convexity' (formerly 'signed tropical convexity') and the novel notion 'TC-convexity'. We derive several separation results for TO-convexity and TC-convexity. A key ingredient is a thorough understanding of TC-hemispaces - those TC-convex sets whose complement is also TC-convex. Furthermore, we use new insights in the interplay between convexity over Puiseux series and its signed valuation. Remarkably, TC-convexity can be seen as a natural convexity notion for representing oriented matroids as it arises from a generalization of the composition operation of vectors in an oriented matroid. We make this explicit by giving representations of linear spaces over the real tropical hyperfield in terms of TC-convexity.