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  "body": "\n## Example\n\nLet $n \\ge 1$.\nThe *Birkhoff polytope* $B_n$ is the set of $n \\times n$ [[doubly stochastic matrices|JJA90EB2PS]], seen as points of $\\R^{n \\times n}$:\n$$B_n = \\set{X \\in \\R^{n \\times n} : x_{ij} \\ge 0, \\ \\textstyle\\sum_{j} x_{ij} = 1 \\text{ for all } i, \\ \\sum_{i} x_{ij} = 1 \\text{ for all } j}.$$\nIt is bounded, since $0 \\le x_{ij} \\le 1$, and is cut out by finitely many linear inequalities and equations, so it is a [[convex polytope|300K6CB1KH]].\nBy the [[Birkhoff–von Neumann theorem|3KY1RMEDMJ]], $B_n$ is the convex hull of the $n!$ [[permutation matrices|08SP3DF8NB]], and these are its [[vertices|R7V3S9V891]].\nThe $2n$ equations have rank $2n - 1$, so $\\dim B_n = n^2 - (2n - 1) = (n-1)^2$.\nFor $n \\ge 3$ the [[facets|DPDXN2FJC4]] are the $n^2$ sets $\\set{X \\in B_n : x_{ij} = 0}$.\n\n## Remarks\n\nFor $n = 2$, $B_2$ is the segment of matrices $\\begin{pmatrix} t \u0026 1-t \\\\ 1-t \u0026 t \\end{pmatrix}$, $0 \\le t \\le 1$, with the two permutation matrices as endpoints.\nFor $n = 3$, $B_3$ is a $4$-dimensional polytope with $6$ vertices and $9$ facets.\nIts role in optimisation is that a linear function on $B_n$ attains its maximum at a permutation matrix, which solves the assignment problem by linear programming.\n\n## Explanations\n\n### L1 human\n\nA doubly stochastic matrix is a square table of nonnegative numbers in which every row and every column adds up to $1$.\nAll such tables together form a solid shape, the Birkhoff polytope.\nIts corners are the permutation matrices, tables of zeros with a single $1$ in each row and each column.\nEvery doubly stochastic matrix is therefore an average of permutation matrices.\n",
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