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  "body": "\n## Definition\n\nLet $K$ be a geometric simplicial complex in $\\R^N$, so each simplex $\\sigma$ of $K$ is the [[convex hull|0FQSDC2EQ2]] of affinely independent vertices $v_0, \\dots, v_n$.\nThe *barycentre* of $\\sigma$ is the point\n$$b(\\sigma) = \\frac{1}{n + 1} \\sum_{i=0}^{n} v_i.$$\nThe *barycentric subdivision* $\\operatorname{sd} K$ is the geometric simplicial complex with the following simplices.\nFor each chain $\\sigma_0 \\subsetneq \\sigma_1 \\subsetneq \\cdots \\subsetneq \\sigma_k$ of simplices of $K$, it has the convex hull of $\\set{b(\\sigma_0), \\dots, b(\\sigma_k)}$.\n\n## Remarks\n\nThe union of the simplices of $\\operatorname{sd} K$ equals the union of the simplices of $K$, and each simplex of $\\operatorname{sd} K$ lies in a simplex of $K$.\nIf every simplex of $K$ has diameter at most $d$ and $K$ has dimension at most $n$, then every simplex of $\\operatorname{sd} K$ has diameter at most $\\frac{n}{n + 1} d$.\n\nFor an abstract [[simplicial complex|V8GZ0W1ZZ5]] $K$, the vertices of $\\operatorname{sd} K$ are the nonempty faces of $K$.\nThe faces of $\\operatorname{sd} K$ are the finite chains of nonempty faces under inclusion.\n\nExample: let $K$ be a $2$-simplex with vertices $v_0, v_1, v_2$ and all its faces.\nThe maximal chains are $\\set{v_i} \\subsetneq \\set{v_i, v_j} \\subsetneq \\set{v_0, v_1, v_2}$ for the $6$ ordered pairs $i \\ne j$, so $\\operatorname{sd} K$ has $6$ triangles, $7$ vertices and $12$ edges.\n\nNon-example: cutting the triangle into $3$ triangles by joining its centroid to the three corners gives a subdivision, but not the barycentric one.\nThe barycentres of the edges are not vertices of it, though each edge is a simplex of $K$.\n",
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