Biconvex Polytopes and Tropical Linear Spaces

TL;DR

This paper establishes a connection between biconvex polytopes and tropical linear spaces by constructing matroid subdivisions of the hypersimplex, providing explicit subdivision guidelines without computational tools.

Contribution

It introduces a novel construction linking biconvex polytopes to tropical linear spaces via matroid subdivisions, with manual subdivision methods demonstrated in rank-4 cases.

Findings

Biconvex polytopes are shown to be cells of tropical linear spaces.

A manual subdivision method for the hypersimplex is developed.

An injection from vertices of biconvex polytopes to monomials is established.

Abstract

A biconvex polytope is a classical and tropical convex hull of finitely many points. Given a biconvex polytope, for each vertex of it we construct a directed bigraph and a gammoid so that the collection of base polytopes of those gammoids is a matroid subdivision of the hypersimplex, thereby proving a biconvex polytope arises as a cell of a tropical linear space. Our construction provides manually feasible guidelines for subdividing the hypersimplex into base polytopes, without resorting to computers. We work out the rank-4 case as a demonstration. We also show there is an injection from the vertices of any (k-1)-dimensional biconvex polytope into the degree-(k-1) monomials in k indeterminates.