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Tropical balls, geodesics and honeycomb

Amnon Rosenmann

TL;DR

This paper investigates geometry under the tropical metric $d_{tr}$ on the tropical projective torus, defining tropical length and geodesics and classifying compact trop geodesic sets as polytope intersections of the form $a_i\le x_i\le a_i'$ and $x_i-x_j\ge b_{ij}$. Central to the study is the tropical unit ball $B_{tr}^n$, shown to admit multiple equivalent descriptions: as a zonotope via the Minkowski sum of $n{+}1$ tropical unit segments, as a union of $n{+}1$ tropical unit hypercubes, as the tropical geodesic hull of the unit vectors, and as the tropical convex hull of their negations. The paper also defines a tropical angle on the unit sphere and develops a constructive proof that translates of $B_{tr}^n$ with centers on a sublattice tile $\mathbb{R}^n$ facet-to-facet, forming a tropical honeycomb. Together, these results connect tropical convexity, zonotopal geometry, and space tilings, with implications for tropical geometry and related optimization and combinatorial structures.

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 $\mathbb{R}^n$.

Tropical balls, geodesics and honeycomb

TL;DR

This paper investigates geometry under the tropical metric on the tropical projective torus, defining tropical length and geodesics and classifying compact trop geodesic sets as polytope intersections of the form and . Central to the study is the tropical unit ball , shown to admit multiple equivalent descriptions: as a zonotope via the Minkowski sum of tropical unit segments, as a union of tropical unit hypercubes, as the tropical geodesic hull of the unit vectors, and as the tropical convex hull of their negations. The paper also defines a tropical angle on the unit sphere and develops a constructive proof that translates of with centers on a sublattice tile facet-to-facet, forming a tropical honeycomb. Together, these results connect tropical convexity, zonotopal geometry, and space tilings, with implications for tropical geometry and related optimization and combinatorial structures.

Abstract

We study the geometry of tropical balls in 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 and then characterize compact tropically geodesic sets in . 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 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 form a facet-to-facet honeycomb tiling of .
Paper Structure (14 sections, 11 theorems, 34 equations, 6 figures, 1 table)

This paper contains 14 sections, 11 theorems, 34 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The tropical projective torus
  3. Tropical metric
  4. Tropical geodesics
  5. Compact tropically geodesic sets in the plane
  6. Compact tropically geodesic sets in $\mathbb{R}^{n}$
  7. The tropical unit ball
  8. Tropical coordinate system
  9. Minkowski sum
  10. The tropical angle between two (standard) lines
  11. Decomposition into tropical unit hypercubes
  12. Tropical geodesic hull
  13. Tropical linear combination and tropical convexity
  14. Honeycomb of $\mathbb{R}^{n}$ with tropical balls

Key Result

Proposition 4.7

The following are equivalent characterizations of $g_{\mathrm{tr}}(\{{\bf x}, {\bf y} \})$.

Figures (6)

  • Figure 1: Tropical circumference of a standard circle
  • Figure 2: (a) Min tropical line segment; (b) max tropical line segment; (c) tropical geodesic hull of $\{{\bf p},{\bf q}\}$
  • Figure 3: Compact tropically geodesic 2D sets in $\mathbb{R}^2$
  • Figure 4: The tropical unit ball in the plane (a hexagon, left) and in space (a dodecahedron with four-sided faces, right)
  • Figure 5: Decomposition of the 2-dimensional tropical unit ball into tropical unit hypercubes: min-plus decomposition (left) and max-plus decomposition (right)
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 4.1
  • Remark 4.2
  • Example 4.3
  • Definition 4.4
  • Proposition 4.7
  • Definition 4.8
  • ...and 30 more