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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *proper cone* in $\\R^n$ is a [[convex cone|6VE4KA12VD]] $K \\subseteq \\R^n$ such that:\n\n1. $K$ is nonempty and [[closed|0ZCK8DQYN1]];\n2. $K$ has nonempty [[interior|0RZH6W98HK]];\n3. $K$ is salient: $K \\cap (-K) = \\set{0}$, so $K$ contains no line through the origin.\n\n## Remarks\n\nA nonempty closed convex cone contains $0 = \\lim_{t \\to 0^+} t x$, so a proper cone is a [[pointed cone|J9HA54ACME]].\n\nExample: the nonnegative orthant $\\R^n_{\\ge 0}$.\nIt is a closed convex cone, its interior is the set of vectors with all coordinates positive, and if $x$ and $-x$ both have nonnegative coordinates then $x = 0$.\n\nNon-example: the closed half-plane $\\set{(x, y) \\in \\R^2 : y \\ge 0}$ satisfies clauses 1 and 2, but it contains $(1, 0)$ and $(-1, 0)$, so clause 3 fails.\n\nA proper cone $K$ gives a partial order on $\\R^n$ by $x \\preceq_K y$ if $y - x \\in K$.\nConventions vary: some sources, Mathlib among them, call any closed pointed cone proper and drop clauses 2 and 3.\n",
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