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Fundamentals of Broken Line Convex Geometry

Juan Bosco Frías-Medina, Timothy Magee

TL;DR

This work extends convex geometry to the rational tropicalization of cluster varieties by introducing broken line convexity, where broken line segments replace straight lines. The authors develop a comprehensive framework including the tropical Minkowski sum $+_{\vartheta}$, the broken line convex hull $\mathrm{conv}_{BL}$, and a dual polyhedral theory with tropical half-spaces, faces, and polar duality, all tailored to the cluster-analytic setting via $\vartheta$-functions and scattering diagrams. They prove broken line analogues of classical results, establish the equivalence between GHKK's min-convex and decreasing notions, and show compatibility between $\mathrm{conv}_{BL}$ and tropical Minkowski sums. The polyhedral side features the weak face fan, duality of faces with polar sets, and robust vertex/half-space representations, laying groundwork for a Borisov-type duality program in the study of minimal models for cluster varieties.

Abstract

We develop the fundamentals of a new theory of convex geometry -- which we call "broken line convex geometry". This is a theory of convexity where the ambient space is the rational tropicalization of a cluster variety, as opposed to an ambient vector space. In this theory, line segments are replaced by broken line segments, and we adopt the notion of convexity in [CMN21]. We state and prove broken line convex geometry versions of many standard results from usual convex geometry.

Fundamentals of Broken Line Convex Geometry

TL;DR

This work extends convex geometry to the rational tropicalization of cluster varieties by introducing broken line convexity, where broken line segments replace straight lines. The authors develop a comprehensive framework including the tropical Minkowski sum , the broken line convex hull , and a dual polyhedral theory with tropical half-spaces, faces, and polar duality, all tailored to the cluster-analytic setting via -functions and scattering diagrams. They prove broken line analogues of classical results, establish the equivalence between GHKK's min-convex and decreasing notions, and show compatibility between and tropical Minkowski sums. The polyhedral side features the weak face fan, duality of faces with polar sets, and robust vertex/half-space representations, laying groundwork for a Borisov-type duality program in the study of minimal models for cluster varieties.

Abstract

We develop the fundamentals of a new theory of convex geometry -- which we call "broken line convex geometry". This is a theory of convexity where the ambient space is the rational tropicalization of a cluster variety, as opposed to an ambient vector space. In this theory, line segments are replaced by broken line segments, and we adopt the notion of convexity in [CMN21]. We state and prove broken line convex geometry versions of many standard results from usual convex geometry.
Paper Structure (14 sections, 58 theorems, 164 equations, 6 figures)

This paper contains 14 sections, 58 theorems, 164 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results from CMMM
  4. Scattering diagrams and broken lines
  5. Tropical Minkowski sum
  6. Convexity for functions on tropical spaces
  7. Definition and characterization
  8. Equivalence of GHKK's "min-convex" and "decreasing"
  9. Basic results
  10. Broken line convex polyhedral geometry
  11. Faces
  12. The weak face fan
  13. Duality for polyhedral sets and faces
  14. Half-space and vertex representations

Key Result

Theorem 2

Let $S\subset U^{\mathrm{trop} }(\mathbb{Q} )$ be broken line convex. Then $\varphi:S \to \mathbb{Q}$ is convex with respect to broken lines if and only if for all $s_1,\dots, s_d, s \in S$, $a_1, \dots, a_d \in \mathbb{Q} _{\geq 0}$ with $a_1 s_1,\dots, a_d s_d$, and $(a_1+\cdots +a_d)s$ all integr

Figures (6)

  • Figure 1: The tropical Minkowski sum of two points in $(\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ for the $\mathcal{A}$ cluster variety of type $A_2$. As is standard, to draw this picture we identify $(\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ with $\mathbb{Q} ^2$ via a choice of seed. The relevant broken lines appear on the left and the corresponding tropical Minkowski sum on the right.
  • Figure 2: A polytopal set ${\textcolor{blue-ish}{S}}\subset (\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ for the $\mathcal{A}$ cluster variety of type $A_2$. The indicated face $F$ is not broken line convex. As is standard, to draw this picture we identify $(\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ with $\mathbb{Q} ^2$ via a choice of seed.
  • Figure 3: A bigon $S$ in $(\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ together with its faces $\mathcal{F}_{{\textcolor{blue-ish}{S}}}$ for the $\mathcal{A}$ cluster variety of type $A_2$. Note that the intersection of the facets is a pair of vertices rather than a single face.
  • Figure 4: Schematic of $\left.\gamma\right|_{[\tau-\epsilon,\tau+\epsilon]}$ and $\gamma'$ as detailed above.
  • Figure 5: On the left, the bigon ${\textcolor{blue-ish}{S}}$ of Figure \ref{['fig:Bigon1']}. On the right, the weak face fan $\Sigma[{\textcolor{blue-ish}{S}}]$. Note that for $F$ either facet, $\sigma_F$ is only weakly convex, not broken line convex. In fact, $\mathop{\mathrm{conv_{BL}}}\nolimits\left(\sigma_F\right)=(\mathcal{A} ^\vee)^{\mathrm{trop} }(\mathbb{Q} )$ for both facets.
  • ...and 1 more figures

Theorems & Definitions (142)

  • Definition 1: \ref{['def:ConvWRTBL']}
  • Theorem 2: \ref{['prop:ConvWRTBL']}, \ref{['rem:ConvWRTBL-Equiv']}
  • Definition 3: \ref{['def:Half-space-Hyperplane']}
  • Definition 4: BLC
  • Definition 5: GHKK
  • Definition 6: BLC
  • Theorem 8: ThetaReciprocity,"Theta Reciprocity"
  • Theorem 9: ThetaReciprocity,"Valuative Independence"
  • Definition 11
  • Definition 12
  • ...and 132 more