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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|05EGHEQZAW]].\nA *positive cone* in $V$ is a subset $P \\subseteq V$ such that:\n\n1. $x + y \\in P$ for all $x, y \\in P$;\n2. $\\lambda x \\in P$ for all $x \\in P$ and all real $\\lambda \\ge 0$;\n3. $P \\cap (-P) = \\set{0}$.\n\n## Remarks\n\nA positive cone $P$ defines a [[partial order|011C29Q1S1]] on $V$ by $x \\le y \\iff y - x \\in P$.\nThis order satisfies $x \\le y \\implies x + z \\le y + z$ and $x \\le y \\implies \\lambda x \\le \\lambda y$ for $\\lambda \\ge 0$, and $P = \\set{x : 0 \\le x}$.\nConversely, for each partial order with these two properties, the set $\\set{x : 0 \\le x}$ is a positive cone.\nExample: $P = \\set{x \\in \\R^n : x_i \\ge 0 \\text{ for all } i}$ is a positive cone, and it gives the componentwise order.\nExample: the [[positive semidefinite matrices|0880F9EQCP]] form a positive cone in the space of real symmetric $n \\times n$ matrices.\nNon-example: the half-plane $\\set{(x, y) \\in \\R^2 : y \\ge 0}$ satisfies clauses 1 and 2, but contains $(1, 0)$ and $(-1, 0)$, so it fails clause 3.\nSome authors drop clause 3 and call $P$ a *wedge* or a convex cone; then the relation $\\le$ is only a preorder.\n",
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