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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]].\nA subset $C \\subseteq V$ is a *convex cone* if for all $x, y \\in C$ and all real numbers $\\alpha \u003e 0$ and $\\beta \u003e 0$,\n$$\\alpha x + \\beta y \\in C.$$\n\n## Remarks\n\nEquivalently, $C$ is closed under addition and under multiplication by positive scalars.\nEquivalently again, $C$ is [[convex|02AS45T3H9]] and $t x \\in C$ for all $x \\in C$ and $t \u003e 0$.\n\nExample: the nonnegative orthant $\\R^n_{\\ge 0} = \\set{x \\in \\R^n : x_i \\ge 0 \\text{ for all } i}$.\nIf $x, y$ have nonnegative coordinates and $\\alpha, \\beta \u003e 0$, then each coordinate $\\alpha x_i + \\beta y_i$ is nonnegative.\n\nNon-example: the closed unit disc in $\\R^2$ is convex, but $(1, 0)$ lies in it and $2 \\cdot (1, 0) = (2, 0)$ does not.\n\nConventions differ: many authors also allow $\\alpha = \\beta = 0$, so that every nonempty convex cone contains $0$.\nA convex cone that contains $0$ is called a pointed cone in some texts.\nThe same definition works over any ordered field in place of $\\R$.\n",
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