Tropical balls, geodesics and honeycomb

Abstract

In these notes we describe the geometry of tropical balls in $\mathbb{R}^n$ equipped with the tropical metric. After defining the tropical length of rectifiable curves (and not just piecewise linear curves), we characterize compact tropically geodesic sets in $\mathbb{R}^n$. Next, we describe the tropical unit ball as a zonotope, 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 show 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 $\mathbb{R}^n$. We note that a great part of the material presented here is either known or implied from known results.