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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $P \\subseteq \\R^n$ be a [[convex polytope|300K6CB1KH]].\n\n1. A *V-representation* of $P$ is a finite set $V \\subseteq \\R^n$ with $P = \\mathrm{conv}(V)$, the [[convex hull|0FQSDC2EQ2]] of $V$.\n2. An *H-representation* of $P$ is a pair $(A, b)$ of a real $m \\times n$ matrix $A$ and a vector $b \\in \\R^m$ with $P = \\set{x \\in \\R^n : A x \\le b}$. Here $\\le$ is taken coordinatewise.\n\nEquivalently, an H-representation is a finite family of closed half-spaces whose intersection is $P$.\n\n## Remarks\n\nExample: for $P = [0, 1]^2$, the set $V = \\set{(0, 0), (1, 0), (0, 1), (1, 1)}$ is a V-representation, and the inequalities $-x_1 \\le 0$, $x_1 \\le 1$, $-x_2 \\le 0$, $x_2 \\le 1$ form an H-representation.\nNon-example: the inequalities $-x_1 \\le 0$, $-x_2 \\le 0$ alone are not an H-representation of $[0, 1]^2$; they define the unbounded quadrant, which contains $(2, 0) \\notin P$.\nRepresentations are not unique, since points and inequalities can be added without changing $P$.\nThe smallest V-representation is the set of vertices of $P$, and for a full-dimensional $P$ an irredundant H-representation has exactly one inequality for each facet, unique up to positive scaling.\nEvery convex polytope has both kinds of representation; this is the Minkowski–Weyl theorem.\n",
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