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  "body": "\n## Definition\n\nA $k$-simplex is the [[convex hull|0FQSDC2EQ2]] of $k + 1$ affinely independent points.\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 1$.\n$P$ is *simplicial* if every [[facet|DPDXN2FJC4]] of $P$ is a $(d-1)$-simplex.\n\n## Remarks\n\nEquivalently, every proper face of $P$ is a simplex, since each face of a simplex is a simplex.\nEquivalently again, every facet of $P$ has exactly $d$ vertices.\n\nExample: the octahedron $\\operatorname{conv}\\set{\\pm e_1, \\pm e_2, \\pm e_3}$ in $\\R^3$ is simplicial.\nIts facets are the eight triangles $\\operatorname{conv}\\set{\\epsilon_1 e_1, \\epsilon_2 e_2, \\epsilon_3 e_3}$ with $\\epsilon_i \\in \\set{1, -1}$, and each is the hull of $3$ affinely independent points.\n\nNon-example: the cube $[0,1]^3$ is not simplicial, because its facets are squares with $4 \u003e 3$ vertices.\n\nEvery polygon is simplicial, since its facets are segments.\nThe convex hull of finitely many points in general position in $\\R^d$, with at least $d+1$ of them, is simplicial.\n",
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