# Tropical bisectors and Voronoi diagrams

## TL;DR

This paper introduces the concept of tropical Voronoi diagrams, exploring their structure and algorithms by studying bisectors under polyhedral norms, with a focus on tropical geometry.

## Contribution

It initiates the study of tropical Voronoi diagrams, providing structural insights and algorithms based on bisectors of points with respect to polyhedral norms.

## Key findings

- Bisectors of three points are homeomorphic to open subsets of Euclidean space under certain conditions.
- Structural properties of tropical bisectors are established.
- Algorithms for constructing tropical Voronoi diagrams are proposed.

## 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.