Hypersimplex information

Standard hypersimplices in $\mathbb {R} ^{3}$
$\Delta _{3,1}$ Hyperplane: $x+y+z=1$	$\Delta _{3,2}$ Hyperplane: $x+y+z=2$

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 $0<k<d$ .^[1]

^ Miller, Ezra; Reiner, Victor; Sturmfels, Bernd, Geometric Combinatorics, IAS/Park City mathematics series, vol. 13, American Mathematical Society, p. 655, ISBN 9780821886953.

[gc-1] Miller, Ezra; Reiner, Victor; Sturmfels, Bernd, Geometric Combinatorics, IAS/Park City mathematics series, vol. 13, American Mathematical Society, p. 655, ISBN 9780821886953.

Hypersimplex information

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