Tropical linear spaces and tropical convexity

TL;DR

This paper establishes a fundamental equivalence between tropical linear spaces and tropical varieties supported on valuated matroids, and proves a local-to-global convexity principle in tropical geometry.

Contribution

It proves that tropical convexity characterizes tropical linear spaces and introduces a local-to-global convexity principle for tropical sets.

Findings

Tropical varieties supported on valuated matroids are exactly the tropically convex ones.

Any closed, connected, locally tropically convex set is globally tropically convex.

The paper bridges tropical convexity with classical linear space concepts.

Abstract

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tropically convex set is tropically convex.