The $(k,n)$-hypersimplex is a convex polytope $\Delta(k, n)$ (or $\Delta_{k,n}$) in $\mathbb{R}^n$ which is the convex hull of the $\left(\array{n \\ k}\right)$ points $e_{i_1}+ e_{i_2}+ \cdots + e_{i_k}$, $1\leq i_1\lt i_2\lt \ldots \lt i_k\leq n$, where the $e_{j}$ are the standard basis vectors. $\Delta(k,n)$ may be seen as a convex hull of the barycenters of the $(k-1)$-dimensional faces of $a (n-1)$-dimensional simplex. Special cases, $\Delta(1,n)$ and $\Delta(n-1,n)$ are themselves simplices of dimension $(n-1)$.
The combinatorics of hypersimplices extends the combinatorics of distinguished triangles and octahedra in the standard triangulated categories; in fact they are postulated in Maltsiniotis’s strong version of a triangulated category. The octahedron is a $(2,4)$-hypersimplex. Higher hypersimplices were indeed obtained from $A_\infty$-enrichments by Volodymyr Lyubashenko, so one could expect that they can also be obtained from the homotopy category of a stable (∞,1)-category. There are also connections between distinguished hypersimplices and Postnikov towers in triangulated categories.
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