Halfspace Representations of Path Polytopes of Trees

TL;DR

This paper provides a minimal halfspace representation of path polytopes of trees, constructed inductively via toric fiber products, with applications in phylogenetics, tropical geometry, and algebraic statistics.

Contribution

It introduces a novel inductive method to derive minimal halfspace representations of path polytopes of trees using toric fiber products.

Findings

Explicit minimal halfspace descriptions for path polytopes.

Inductive construction method via toric fiber products.

Applications demonstrated in phylogenetics and algebraic geometry.

Abstract

Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics, tropical geometry, and algebraic statistics. We provide a minimal halfspace representation of these polytopes. The construction is made inductively using toric fiber products.