  In this paper we present an explicit combinatorial description of a special
class of facets of the secondary polytopes of hypersimplices. These facets
correspond to polytopal subdivisions called multi-splits. We show a relation
between the cells in a multi-split of the hypersimplex and nested matroids.
Moreover, we get a description of all multi-splits of a product of simplices.
Additionally, we present a computational result to derive explicit lower bounds
on the number of facets of secondary polytopes of hypersimplices.

Combinatorics