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  "body": "\n## Summary\n\nPseudoconvexity is the notion of convexity adapted to several complex variables.\nAn open set in $\\C^n$ is pseudoconvex if it admits a continuous [[plurisubharmonic|FF2P7ER3QT]] exhaustion function.\nSuch a function tends to $+\\infty$ at the boundary and, for unbounded sets, at infinity.\nUnlike holomorphic convexity, it is a local condition at the boundary, and for smooth boundaries it can be checked with the Levi form.\nBy the solution of the Levi problem, the pseudoconvex domains in $\\C^n$ are exactly the domains of holomorphy.\n\n## Explanations\n\n### L1 human\n\nOrdinary convexity can be tested with convex functions: a convex region carries a convex function that blows up at its edge.\nPseudoconvexity replaces convex functions by plurisubharmonic ones, which only need to be subharmonic along each complex line.\nThis condition is much weaker than convexity and is preserved by holomorphic changes of coordinates.\nEvery convex open set and every open set in $\\C$ is pseudoconvex, but the punctured ball $\\set{0 \u003c \\abs{z} \u003c 1}$ in $\\C^2$ is not.\n",
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