Hypersimplex
In polyhedral combinatorics, the hypersimplex is a convex polytope that generalizes the simplex. It is determined by two integers and, and is defined as the convex hull of the -dimensional vectors whose coefficients consist of ones and zeros. Equivalently, can be obtained by slicing the -dimensional unit hypercube with the hyperplane of equation and, for this reason, it is a -dimensional polytope when.
Properties
The number of vertices of is. The vertex-edge graph of the hypersimplex is the Johnson graph.Alternative constructions
An alternative construction is to take the convex hull of all -dimensional -vectors that have either or nonzero coordinates. This has the advantage of operating in a space that is the same dimension as the resulting polytope, but the disadvantage that the polytope it produces is less symmetric.The hypersimplex is also the matroid polytope for a uniform matroid with elements and rank.
Examples
The hypersimplex is a -simplex.The hypersimplex is an octahedron, and the hypersimplex is a rectified 5-cell.
Generally, the hypersimplex,, corresponds to a uniform polytope, being the -rectified -dimensional simplex, with vertices positioned at the center of all the -dimensional faces of a -dimensional simplex.