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  "body": "\n## Definition\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 1$.\nFor $0 \\le i \\le d - 1$, let $f_i(P)$ be the number of [[faces|KY7SB854VH]] of $P$ of dimension $i$.\nThe *f-vector* of $P$ is $(f_0(P), f_1(P), \\dots, f_{d-1}(P))$.\n\n## Remarks\n\nSome authors add $f_{-1} = 1$ for the empty face and $f_d = 1$ for $P$ itself.\nFor a finite [[simplicial complex|V8GZ0W1ZZ5]] $\\Delta$, $f_i(\\Delta)$ is the number of $i$-dimensional simplices of $\\Delta$, defined in the same way.\n\nExample: the cube $[0,1]^3$ has $8$ vertices, $12$ edges and $6$ square faces, so its f-vector is $(8, 12, 6)$.\n\nNon-example: $(4, 6, 3)$ is not the f-vector of any $3$-dimensional polytope.\nEvery $3$-polytope satisfies $f_0 - f_1 + f_2 = 2$, while $4 - 6 + 3 = 1$.\n",
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