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Signed tropical halfspaces and convexity

Georg Loho, Mateusz Skomra

Abstract

We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and Végh (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.

Signed tropical halfspaces and convexity

Abstract

We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and Végh (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.
Paper Structure (26 sections, 66 theorems, 159 equations, 10 figures)

This paper contains 26 sections, 66 theorems, 159 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on signed tropical numbers
  3. Halfspaces
  4. Basic calculations in $\mathbb{S}$
  5. Left sum
  6. Flavors of signed convexity
  7. TO-convexity
  8. TC-convexity
  9. Representation of TC-convex sets by combinations
  10. TC-convex cones
  11. Separation by Hemispaces
  12. Separation of TO-convex sets
  13. Separation of TC-convex sets
  14. TC-hemispaces are nearly halfspaces
  15. The boundary of TC-hemispaces
  16. ...and 11 more sections

Key Result

Lemma 2.4

The set $\overline{\mathcal{H}}^+(a)$ is closed, its interior is equal to $\mathcal{H}^+(a)$, and the closure of $\mathcal{H}^+(a)$ is equal to $\overline{\mathcal{H}}^+(a)$.

Figures (10)

  • Figure 1: An open and a closed signed tropical halfspace.
  • Figure 2: TO-convex intervals and TC-convex intervals in the plane (see \ref{['ex:TO+TC-intervals']})
  • Figure 3: The Pasch property in the real plane.
  • Figure 4: The Pasch property in the TO-convexity is satisfied.
  • Figure 5: The Pasch property in the TC-convexity is not satisfied.
  • ...and 5 more figures

Theorems & Definitions (154)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: LohoVegh:2020
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 144 more