Tropical Convex Hull Computations

TL;DR

This survey explores tropical convex hulls, their combinatorial properties, and algorithms, highlighting connections to various mathematical concepts and demonstrating computational tools like polymake for tropical polytope analysis.

Contribution

It provides a comprehensive overview of tropical polytopes, their combinatorial relationships, and practical algorithms, including software implementation details.

Findings

Tropical convexity relates to multiple combinatorial concepts.

Algorithms for tropical polytope computation are discussed.

Software tools like polymake facilitate tropical polytope analysis.

Abstract

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes.