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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *polytope* in $\\R^n$ is a set $P = P_1 \\cup \\cdots \\cup P_k$ that is the union of finitely many [[convex polytopes|300K6CB1KH]] $P_1, \\ldots, P_k$ in $\\R^n$.\n\n## Remarks\n\nExample: the L-shaped region $[0, 2] \\times [0, 1] \\cup [0, 1] \\times [0, 2]$ is the union of two rectangles, which are convex polytopes, so it is a polytope.\nIt is not convex: it contains $(2, 1)$ and $(1, 2)$, but not their midpoint $(3/2, 3/2)$.\nNon-example: the closed unit disc in $\\R^2$ is not a polytope.\nThe boundary of a finite union of convex polygons lies in finitely many lines.\nEach line meets the unit circle in at most two points, so this boundary cannot contain the whole circle.\nConventions differ between authors.\nIn convex geometry and in polyhedral combinatorics \"polytope\" usually means convex polytope, and some authors require a polytope to be a union of simplices that meet in common faces.\n",
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