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  "body": "\n## Example\n\nFor $d \\ge 1$, let $e_1, \\dots, e_d$ be the standard basis of $\\R^d$.\nThe $d$-dimensional *cross-polytope* is\n$$X_d = \\operatorname{conv}\\set{\\pm e_1, \\dots, \\pm e_d} = \\set{x \\in \\R^d : \\abs{x_1} + \\cdots + \\abs{x_d} \\le 1}.$$\nIt is a [[convex polytope|300K6CB1KH]] with the $2d$ [[vertices|R7V3S9V891]] $\\pm e_i$.\n\nA set of vertices spans a proper face exactly when it contains no pair $e_i, -e_i$.\nIndeed, for signs $\\epsilon_i \\in \\set{1, -1}$ the vector $c = (\\epsilon_1, \\dots, \\epsilon_d)$ gives $c \\cdot x \\le 1$ on $X_d$, with equality on the hull of $\\epsilon_1 e_1, \\dots, \\epsilon_d e_d$.\nSo the [[facets|DPDXN2FJC4]] are the $2^d$ simplices $\\operatorname{conv}\\set{\\epsilon_1 e_1, \\dots, \\epsilon_d e_d}$, and $X_d$ is a [[simplicial polytope|YVJ15PGKDM]].\nChoosing $k + 1$ of the $d$ axes and a sign for each gives the number of $k$-faces:\n$$f_k(X_d) = 2^{k+1} \\binom{d}{k+1}, \\qquad 0 \\le k \\le d-1.$$\n\n## Remarks\n\nFor $d = 2$ the cross-polytope is a square turned by $45$ degrees, and for $d = 3$ it is the regular octahedron.\nIt is the [[polar polytope|6DXF1T506B]] of the [[hypercube|P9PYM71GMM]] $[-1,1]^d$: a vector $y$ satisfies $x \\cdot y \\le 1$ for all $x \\in [-1,1]^d$ exactly when $\\abs{y_1} + \\cdots + \\abs{y_d} \\le 1$.\nIts unit ball description shows that it is the unit ball of the $\\ell^1$ norm.\n\n## Explanations\n\n### L1 human\n\nPut a point at distance one on each side of the origin along every axis, and wrap a skin around these points.\nIn the plane you get a diamond, in space an octahedron.\nThe cross-polytope is the same shape in any dimension.\nIts faces are all triangles or their higher versions, simplices, which makes it the dual of the cube.\n",
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