On the Facets of the Secondary Polytope

TL;DR

This paper explores the facets of the secondary polytope, focusing on splits and introducing k-splits to classify polytopes, with applications to matroid subdivisions and tropical geometry.

Contribution

It extends the study of secondary polytope facets by analyzing splits, introducing k-splits, and applying these concepts to matroid subdivisions and tropical geometry.

Findings

Facets of secondary polytopes include splits and more complex subdivisions.

Introduction of k-splits for classifying polytopes based on facet complexity.

Application of these concepts to matroid subdivisions and tropical geometry.

Abstract

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.