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  "body": "\n## Summary\n\nA proper cone in $\\R^n$ is a closed [[convex cone|6VE4KA12VD]] with nonempty interior that contains no line.\nProper cones define the generalized inequalities $x \\preceq_K y \\iff y - x \\in K$ used in conic and semidefinite optimisation.\nThe nonnegative orthant, the second-order cone and the cone of positive semidefinite matrices are the standard examples.\nThe [[dual cone|QFY1JR1TK4]] of a proper cone is again proper.\n\n## Explanations\n\n### L1 human\n\nA proper cone is a solid, closed wedge with a sharp tip at the origin.\nSolid means it has volume, closed means it contains its boundary, and sharp means it does not contain any full line.\nThe closed first quadrant of the plane is a proper cone.\nA closed half-plane is not, because it contains a full line through the origin.\n",
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