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  "body": "\n## Summary\n\nIn 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|0FQSDC2EQ2]] 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 $0\u003ck\u003cd$.\n",
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