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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]].\nA *pointed cone* in $V$ is a [[convex cone|6VE4KA12VD]] $C \\subseteq V$ with $0 \\in C$.\n\n## Remarks\n\nEquivalently, $C$ is nonempty and $\\alpha x + \\beta y \\in C$ for all $x, y \\in C$ and all real $\\alpha, \\beta \\ge 0$.\nIn this form a pointed cone is a submodule of $V$ over the semiring of nonnegative reals.\n\nExample: the closed upper half-plane $H = \\set{(x, y) \\in \\R^2 : y \\ge 0}$.\nIt is closed under sums and positive multiples, and it contains $(0, 0)$.\nIt contains the whole $x$-axis, so it is pointed in this sense but not salient.\n\nNon-example: the open quadrant $\\set{(x, y) : x \u003e 0, y \u003e 0}$ is a convex cone, but it does not contain $(0, 0)$.\n\nWarning on terminology: many texts on convex analysis say pointed for a convex cone $C$ with $C \\cap (-C) \\subseteq \\set{0}$, that is, a cone that contains no line.\nThat property is called salient here.\n",
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