Fundamentals of Broken Line Convex Geometry
Algebraic Geometry
2026-01-19 v2 Combinatorics
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.
Cite
@article{arxiv.2407.02427,
title = {Fundamentals of Broken Line Convex Geometry},
author = {Juan Bosco Frías-Medina and Timothy Magee},
journal= {arXiv preprint arXiv:2407.02427},
year = {2026}
}
Comments
34 pages, 6 figures. Feedback welcome. Minor corrections and additions to version 1