Hypersimplex

Summary

In polyhedral combinatorics, the hypersimplex $\Delta_{d,k}$ is a convex polytope that generalizes the simplex. It is determined by two integers $d$ and $k$, and is defined as the convex hull of the $d$-dimensional vectors whose coefficients consist of $k$ ones and $d-k$ zeros. Equivalently, $\Delta_{d,k}$ can be obtained by slicing the $d$-dimensional unit hypercube $[0,1]^d$ with the hyperplane of equation $x_1+\cdots+x_d=k$ and, for this reason, it is a $(d-1)$-dimensional polytope when $0<k<d$.