Fundamentals of Broken Line Convex Geometry

TL;DR

This paper introduces a new convex geometry theory called broken line convex geometry, set in the rational tropicalization of cluster varieties, extending classical convexity concepts to a novel algebraic setting.

Contribution

It develops the foundational principles of broken line convex geometry within the context of cluster varieties, generalizing classical convexity results to this new framework.

Findings

Established fundamental properties of broken line convex sets

Proved analogues of standard convex geometry theorems

Extended convexity notions to tropicalized 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.