Tropical convex hulls of polyhedral sets

TL;DR

This paper explores the properties of tropical convex hulls of polyhedral sets, providing vertex descriptions, dimension characterizations, and bounds on tropical curves, advancing the understanding of tropical convexity in various dimensions.

Contribution

It introduces new vertex descriptions, characterizes tropical convexity in multiple dimensions, and establishes bounds on tropical fan degrees, enhancing the theoretical framework of tropical convexity.

Findings

Vertex description of tropical convex hulls of line segments and rays

Tropical and ordinary convex hulls commute in two dimensions

Lower bounds on degrees of tropical fan curves

Abstract

In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %$\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.