Facets of Secondary Polytopes

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions ofA. While the vertices of the secondary polytope – corresponding to the triangulations ofA – 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 most simple example of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of this 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 will be 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.