Splitting Polytopes

TL;DR

The paper introduces the concept of splits of polytopes, showing that any weight function can be uniquely decomposed into split-related functions plus a prime remainder, and explores the structure of split complexes, especially for hypersimplices.

Contribution

It generalizes the decomposition of finite metric spaces to polytopes and describes the split complexes, including their relation to tropical Grassmannians.

Findings

Unique decomposition of weight functions on polytopes.

Complete descriptions of split complexes of all hypersimplices.

Split complexes are subcomplexes of tropical Grassmannians.

Abstract

A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].