Tropical balls, geodesics and honeycomb

Amnon Rosenmann

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arXiv · 2601.14447

Tropical balls, geodesics and honeycomb

Amnon Rosenmann

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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.

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Amnon Rosenmann. 2026-01-20. Tropical balls, geodesics and honeycomb. https://arxiv.org/abs/2601.14447

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