Tropical Bisectors and Voronoi Diagrams

Abstract

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.

Asymmetric tropical distances and power diagrams

“…For instance, the bisectors turn out to be tropically convex and thus contractible. This is very different from the symmetric case, where topologically nontrivial bisectors do occur [15,Example 3]. We prove that, locally, the asymmetric tropical Voronoi regions of a discrete set of sites behave like (possibly unbounded) tropical polyhedra (Proposition 4.1); yet globally this is not true in general.…”

Tropical Ehrhart Theory and Tropical Volume

“…Driven by this motivation, we develop a novel volume notion for tropical convex sets by a thorough investigation of the tropical analog of lattice point counting. This continues the investigation of intrinsic tropical metric properties that started around a tropical isodiametric inequality [16] and tropical Voronoi diagrams [15].…”

Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling

“…To do this we focus on the set of tropical simplices in the tropical simplicial complex of P , ∆ P , determined by the t = n e combinations of vertices in the minimum vertex set of P , V with |V | = n [17]. Definition 3.9 (Tropical Simplicial Complex (See [17])).…”

“…To do this we focus on the set of tropical simplices in the tropical simplicial complex of P , ∆ P , determined by the t = n e combinations of vertices in the minimum vertex set of P , V with |V | = n [17]. Definition 3.9 (Tropical Simplicial Complex (See [17])). Let V be a set of points such that tconv (V ) = P ∈ R e /R1, the collection of tropical polytopes defined by all subsets V ⊆ V where |V | ≤ e and a tropical polytope P = tconv (V ), is known as the tropical simplicial complex, denoted as ∆ P .…”

“…Employing ( 14)-( 16) in each of the previous examples shows that equality holds when using the result from Proposition 3.12. To show that equality does not always hold in (17), we provide the following example of a tropical polytope in R 3 /R1. Example 3.16.…”