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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a finite [[partially ordered set|0QV2JS9RJ3]], and let $\\R^P$ be the real vector space of functions $f \\colon P \\to \\R$.\nThe *order polytope* of $P$ is the set $\\mathcal{O}(P)$ of all $f \\in \\R^P$ such that:\n\n1. $0 \\le f(x) \\le 1$ for all $x \\in P$.\n2. $f(x) \\le f(y)$ for all $x, y \\in P$ with $x \\le y$.\n\n## Remarks\n\nClause 2 says that $f$ is [[order-preserving|0A7KV2WB7A]].\nThe vertices of $\\mathcal{O}(P)$ are the indicator functions of the [[upper sets|ETCE287E2R]] of $P$.\nIts volume is $e(P)/p!$, where $p = \\abs{P}$ and $e(P)$ is the number of [[linear extensions|W8DZ5KGGHN]].\nFor an integer $m \\ge 1$, the dilate $m \\, \\mathcal{O}(P)$ has $\\Omega(P, m + 1)$ points with integer coordinates, where $\\Omega$ is the [[order polynomial|VDBDJZWPEA]].\n\nExample: for the chain $x \u003c y$, $\\mathcal{O}(P)$ is the triangle $0 \\le f(x) \\le f(y) \\le 1$.\nCheck: its vertices $(0, 0)$, $(0, 1)$, $(1, 1)$ are the indicators of the upper sets $\\emptyset$, $\\set{y}$, $\\set{x, y}$, and its area is $1/2 = e(P)/2!$.\n\nNon-example: the point $f(x) = 1$, $f(y) = 0$ satisfies clause 1 but not clause 2, so it is not in $\\mathcal{O}(P)$.\n\nFor an antichain of $p$ elements, $\\mathcal{O}(P)$ is the unit cube $[0, 1]^p$.\n",
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