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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *convex polytope* in $\\R^n$ is a set of the form\n$$P = \\mathrm{conv}\\set{v_1, \\ldots, v_m} = \\set{\\lambda_1 v_1 + \\cdots + \\lambda_m v_m : \\lambda_i \\ge 0, \\ \\lambda_1 + \\cdots + \\lambda_m = 1},$$\nthe [[convex hull|0FQSDC2EQ2]] of finitely many points $v_1, \\ldots, v_m$ of the [[Euclidean space|0XPR1E85W0]] $\\R^n$.\nThe *dimension* of $P$ is the dimension of its affine hull.\n\n## Remarks\n\nExample: the square $[0, 1]^2$ is $\\mathrm{conv}\\set{(0, 0), (1, 0), (0, 1), (1, 1)}$, since every point $(x, y)$ of the square is $(1 - x)(1 - y) (0, 0) + x (1 - y) (1, 0) + (1 - x) y (0, 1) + x y (1, 1)$, with nonnegative weights of sum $1$.\nNon-example: the closed unit disc is compact and convex but not a convex polytope.\nThe convex hull of finitely many points has only finitely many extreme points, while every point of the unit circle is an extreme point of the disc.\nEquivalently, a convex polytope is a bounded set of the form $\\set{x \\in \\R^n : A x \\le b}$ for a real $m \\times n$ matrix $A$ and $b \\in \\R^m$; this equivalence is the Minkowski–Weyl theorem.\nMany authors allow the empty set as a convex polytope, the hull of no points.\n",
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