Hypersimplex

In polyhedral combinatorics, the hypersimplex

is a convex polytope that generalizes the simplex.

It is determined by two integers

, and is defined as the convex hull of the

-dimensional vectors whose coefficients consist of

can be obtained by slicing the

-dimensional unit hypercube

with the hyperplane of equation

and, for this reason, it is a

[1] The number of vertices of

[1] The graph formed by the vertices and edges of the hypersimplex

is the Johnson graph

[2] An alternative construction (for

) is to take the convex hull of all

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 (although combinatorially equivalent to the result of the other construction).

is also the matroid polytope for a uniform matroid with

elements and rank

vertices).

is an octahedron, and the hypersimplex

Generally, the hypersimplex,

, corresponds to a uniform polytope, being the

-dimensional simplex, with vertices positioned at the center of all the

The hypersimplices were first studied and named in the computation of characteristic classes (an important topic in algebraic topology), by Gabrièlov, Gelʹfand & Losik (1975).