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  "body": "\n## Definition\n\nLet $P \\subset \\R^d$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 2$, and let $F$ be a [[facet|DPDXN2FJC4]] of $P$.\nFor each facet $G$, let $H_G$ be the hyperplane spanned by $G$.\nChoose a point $p \\in \\R^d$ such that:\n\n1. $p$ and $P$ lie on opposite sides of $H_F$, strictly;\n2. for every facet $G \\ne F$, $p$ lies strictly on the same side of $H_G$ as the interior of $P$.\n\nFor $x \\in P$, let $\\pi(x)$ be the point where the segment from $p$ to $x$ meets $H_F$.\nThe *Schlegel diagram* of $P$ based at $F$ is the collection of the sets $\\pi(G')$.\nHere $G'$ runs over the [[faces|KY7SB854VH]] of $P$ other than $P$ and $F$.\n\n## Remarks\n\nEach $\\pi(G')$ is a convex polytope contained in $F$, of the same dimension as $G'$.\nThese polytopes cover $F$, two of them meet in a common face, and their inclusions are those of the faces of $P$.\nSo the diagram is a polytopal subdivision of $F$ that records the whole face lattice of $P$.\nPoints $p$ as in the definition exist; any point beyond $F$ close enough to an interior point of $F$ works.\n\nExample: let $P = [0,1]^3$, $F$ the facet $x_3 = 1$, and $p = (\\tfrac12, \\tfrac12, 1 + \\epsilon)$ with $\\epsilon \u003e 0$ small.\nThen $p$ is above $x_3 = 1$, and on the inner side of the four side planes and of $x_3 = 0$.\nThe bottom square projects to a smaller square in the middle of $F$, and the four side squares project to trapezoids between the two squares.\n\nNon-example: in the same example, $p = (5, \\tfrac12, 2)$ satisfies condition 1 but fails condition 2 for the facet $x_1 = 1$, since $p$ has $x_1 \u003e 1$.\nThen the images are not even inside $F$: the segment from $p$ to the vertex $(1, 0, 0)$ meets the plane $x_3 = 1$ at $(3, \\tfrac14, 1) \\notin F$.\n",
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