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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]].\nThen $D$ is *pseudoconvex* if there is a continuous [[plurisubharmonic function|FF2P7ER3QT]] $\\varphi \\colon D \\to \\R$ with the following property.\nFor every $c \\in \\R$, the set $\\set{z \\in D : \\varphi(z) \u003c c}$ has [[compact|0R6EWFKRCF]] closure contained in $D$.\n\nA function $\\varphi$ with this last property is called an exhaustion function of $D$.\n\n## Remarks\n\nExample: the unit ball $B = \\set{z \\in \\C^n : \\abs{z} \u003c 1}$ with $\\varphi(z) = -\\log(1 - \\abs{z}^2)$.\nThe function $\\abs{z}^2$ is plurisubharmonic, and $t \\mapsto -\\log(1 - t)$ is convex and increasing on $[0, 1)$, so $\\varphi$ is plurisubharmonic and continuous.\nThe set $\\set{\\varphi \u003c c}$ is the ball of radius $\\sqrt{1 - e^{-c}}$ (empty for $c \\le 0$), whose closure is compact and contained in $B$.\n\nNon-example: $D = \\set{z \\in \\C^2 : 0 \u003c \\abs{z} \u003c 1}$.\nLet $\\varphi$ be any continuous plurisubharmonic function on $D$, and let $M$ be its maximum on the compact set $\\set{\\abs{z_1} \\le 1/4, \\ \\abs{z_2} = 1/2} \\subseteq D$.\nFor $0 \u003c \\abs{t} \\le 1/4$, the disc $\\zeta \\mapsto (t, \\zeta/2)$, $\\abs{\\zeta} \\le 1$, lies in $D$, so by subharmonicity $\\varphi(t, 0) \\le M$.\nThe points $(t, 0)$ accumulate at the origin, which is not in $D$, so $\\set{\\varphi \u003c M + 1}$ does not have compact closure in $D$, and no exhaustion of the required kind exists.\n\nEquivalent forms: for $D \\ne \\C^n$, $D$ is pseudoconvex exactly when $-\\log d(z)$ is plurisubharmonic on $D$, where $d(z)$ is the distance from $z$ to the boundary.\nIf $D$ has $C^2$ boundary, pseudoconvexity is equivalent to Levi pseudoconvexity: the Levi form of a defining function is positive semidefinite on the complex tangent space at each boundary point.\n",
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