Tropical balls, geodesics and honeycomb

TL;DR

This paper explores the geometry of tropical balls in n-dimensional space with the tropical metric, characterizes tropical geodesics and sets, and demonstrates a honeycomb tiling formed by their translates.

Contribution

It provides new characterizations of tropical balls, defines tropical geodesics, and proves a honeycomb tiling structure in tropical geometry.

Findings

Characterization of tropical unit ball as a zonotope and union of hypercubes

Explicit description of tropical geodesics and geodesic sets

Proof of honeycomb tiling by translates of tropical unit balls

Abstract

We study the geometry of tropical balls in $\mathbb{R}^n$ equipped with the tropical metric introduced by Cohen, Gaubert and Quadrat, an additive form of Hilbert's projective metric. After defining the tropical length of rectifiable curves, we formulate tropical geodesics in $\mathbb{R}^n$ and then characterize compact tropically geodesic sets in $\mathbb{R}^n$. Next, we present several equivalent descriptions of the tropical unit ball: as a zonotope (Minkowski sum of tropical unit segments), via its tropical generating set, as a union of $n+1$ tropical unit hypercubes, and as the tropical geodesic hull of the tropical unit vectors. Finally, we give an explicit proof that translates of the tropical unit ball whose centers lie in a sublattice of $\mathbb{Z}^n$ form a facet-to-facet honeycomb tiling of $\mathbb{R}^n$.