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  "body": "\n## Summary\n\nThe dual cone of a set $C$ in $\\R^n$ is the set of vectors that make a nonnegative [[dot product|0S52TNNTH9]] with every vector of $C$.\nIt is always a closed [[convex cone|6VE4KA12VD]], and for a closed convex cone $C$ the dual of the dual is $C$ again.\nDual cones are the geometric side of duality in linear and conic optimisation, and they appear in Farkas-type theorems of the alternative.\n\n## Explanations\n\n### L1 human\n\nFor each vector $x$ of $C$, the vectors $y$ with $x \\cdot y \\ge 0$ form a closed half-space whose boundary passes through the origin.\nThe dual cone is the intersection of all these half-spaces.\nSo it collects the directions that make an angle of at most $90°$ with every vector of $C$.\nFor the closed first quadrant of the plane, the dual cone is the first quadrant itself.\nFor a single ray in direction $(1, 0)$, it is the right half-plane $\\set{(x, y) : x \\ge 0}$.\n",
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