Convexity of tropical polytopes

TL;DR

This paper explores the relationships between different notions of convexity in tropical and Euclidean spaces, introducing a construction that links max-plus and min-plus convex sets via a canonical matrix called the dominator.

Contribution

It establishes that sets which are two types of tropical polytopes and Euclidean polytopes simultaneously must be the third, answering an open question and characterizing tropical polytopes as polytropes.

Findings

Max-plus, min-plus, and Euclidean convexity are deeply interconnected.

Sets that are two types of tropical and Euclidean polytopes are necessarily the third.

Row spaces of tropical Kleene star matrices correspond exactly to polytropes.

Abstract

We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of $\mathbb{R}^n$. We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovic and show that row spaces of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig and Kulas.