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  "body": "\n## Example\n\nLet $n \\ge 1$.\nThe *permutohedron* $\\Pi_n$ is the convex hull in $\\R^n$ of the $n!$ points\n$$(\\sigma(1), \\sigma(2), \\dots, \\sigma(n)),$$\nwhere $\\sigma$ runs over all [[permutations|05RWFKVTJ7]] of $\\set{1, \\dots, n}$.\n\nAll these points satisfy $x_1 + \\cdots + x_n = n(n+1)/2$, so $\\Pi_n$ lies in a hyperplane, and it has dimension $n - 1$.\nEach of the $n!$ points is a [[vertex|R7V3S9V891]]: the vector $c = (\\sigma(1), \\dots, \\sigma(n))$ is maximised over the points uniquely at that point, by the rearrangement inequality.\nThe [[facets|DPDXN2FJC4]] correspond to the nonempty proper subsets $S \\subset \\set{1, \\dots, n}$, with facet inequality\n$$\\sum_{i \\in S} x_i \\ge \\frac{\\abs{S}(\\abs{S}+1)}{2},$$\nso there are $2^n - 2$ facets.\nTwo vertices are joined by an edge exactly when one is obtained from the other by swapping the values $k$ and $k+1$ for some $k$.\nHence every vertex lies in $n - 1$ edges and $\\Pi_n$ is a [[simple polytope|7E6FXTGH2Y]].\n\n$\\Pi_n$ is a [[zonotope|Z5SA6A67KF]]: it is a translate of the Minkowski sum of the segments $[0, e_j - e_i]$ for $1 \\le i \u003c j \\le n$.\n\n## Remarks\n\nFor $n = 3$, $\\Pi_3$ is a regular hexagon with vertices the six permutations of $(1,2,3)$.\nFor $n = 4$, $\\Pi_4$ is the truncated octahedron, with $24$ vertices, $14$ facets ($6$ squares and $8$ hexagons) and $36$ edges.\n\n## Explanations\n\n### L1 human\n\nWrite the numbers $1$ to $n$ in every possible order and treat each order as a point in space.\nThe solid spanned by these points is the permutohedron.\nIts corners are the orderings, and walking along an edge means swapping two neighbouring values.\nFor three numbers it is a hexagon, for four numbers a solid with $24$ corners built from squares and hexagons.\n",
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