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  "body": "\n## Definition\n\nLet $d \\ge 2$ and $n \u003e d$ be integers.\nThe *moment curve* in $\\R^d$ is the map $x \\colon \\R \\to \\R^d$, $x(t) = (t, t^2, \\dots, t^d)$.\nLet $t_1 \u003c t_2 \u003c \\cdots \u003c t_n$ be real numbers.\nThe *cyclic polytope* $C(n, d)$ is the [[convex hull|0FQSDC2EQ2]] of the points $x(t_1), \\dots, x(t_n)$.\n\n## Remarks\n\n$C(n,d)$ is a $d$-dimensional [[convex polytope|300K6CB1KH]] whose vertices are all $n$ points $x(t_i)$.\nIts face lattice does not depend on the choice of $t_1 \u003c \\cdots \u003c t_n$, so $C(n, d)$ names a well-defined combinatorial type.\nGale's evenness condition describes its [[facets|DPDXN2FJC4]].\nA set $S$ of $d$ indices gives a facet if and only if, for any two indices $i \u003c j$ outside $S$, an even number of elements of $S$ lie between them.\n\nExample: for $d = 2$ the moment curve is the parabola $y = x^2$, and $C(n, 2)$ is a convex $n$-gon.\nIts facets are the edges $\\set{i, i+1}$ and $\\set{1, n}$, and these are exactly the pairs passing the evenness condition.\n\nNon-example: the regular octahedron is not of the form $C(6, 3)$.\nIn $C(6,3)$ the vertex $x(t_1)$ lies in $5$ edges, while every vertex of the octahedron lies in $4$ edges.\n",
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