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  "body": "\n## Definition\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 1$, where the dimension of a set is the dimension of its [[affine span|0YMB1676R1]].\nA *facet* of $P$ is a [[face|KY7SB854VH]] $F$ of $P$ with $\\dim F = d - 1$.\n\n## Remarks\n\nExample: the cube $[0,1]^3$ has dimension $3$.\nThe square $\\set{x \\in [0,1]^3 : x_3 = 1}$ is the intersection of the cube with the supporting plane $x_3 = 1$, and its affine span is a plane, so it is a facet.\nThe cube has six facets, one for each equation $x_i = 0$ or $x_i = 1$.\n\nNon-example: the edge $\\set{(t, 0, 0) : 0 \\le t \\le 1}$ of the same cube is a face, but its dimension is $1 \\ne 2$, so it is not a facet.\n\nEquivalently, when $P \\subset \\R^d$ is full-dimensional, $F$ is a facet if and only if $F = P \\cap H$ for a supporting hyperplane $H$ such that $F$ spans $H$ affinely.\nA full-dimensional polytope has an irredundant description by linear inequalities, unique up to positive scaling of each inequality.\nIts inequalities correspond one to one to its facets.\nThe faces of dimension $0$ and $1$ are the vertices and edges; for $d = 3$ the facets are the usual two-dimensional faces.\n",
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    "name": "Facet",
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  "name": "Facet",
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