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  "body": "\n## Definition\n\nLet $n \\ge 0$ and $N \\ge n$ be integers.\nLet $v_0, \\ldots, v_n$ be [[affinely independent|JWK49HMHYT]] points of $\\R^N$.\nThe *$n$-simplex* with vertices $v_0, \\ldots, v_n$ is their [[convex hull|0FQSDC2EQ2]]\n$$\\Delta = \\set{t_0 v_0 + \\cdots + t_n v_n : t_0, \\ldots, t_n \\ge 0, \\ t_0 + \\cdots + t_n = 1}.$$\nThe number $n$ is its dimension.\nThe convex hull of a nonempty subset of the vertices is a *face* of $\\Delta$.\n\n## Remarks\n\nThe numbers $t_0, \\ldots, t_n$ are the barycentric coordinates of a point of $\\Delta$, and they are unique by affine independence.\n\nExample: $v_0 = (0, 0)$, $v_1 = (1, 0)$ and $v_2 = (0, 1)$ are affinely independent, since $v_1 - v_0$ and $v_2 - v_0$ are linearly independent.\nTheir $2$-simplex is the triangle $\\set{(x, y) : x, y \\ge 0, \\ x + y \\le 1}$.\n\nNon-example: the points $(0, 0)$, $(1, 0)$ and $(2, 0)$ are not affinely independent.\nTheir convex hull is the segment from $(0, 0)$ to $(2, 0)$, a $1$-simplex but not a $2$-simplex.\n\nThe *standard $n$-simplex* is the simplex in $\\R^{n+1}$ with the standard basis vectors as vertices.\nSome texts, and Mathlib, use the word simplex for the family of vertices rather than for their convex hull.\n",
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