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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a domain with $C^2$ boundary, and let $\\rho$ be a defining function of $\\Omega$.\nFor $p \\in \\partial\\Omega$, let $H_p$ be the complex tangent space of $\\partial\\Omega$ at $p$.\nThe domain $\\Omega$ is *Levi pseudoconvex* if the [[Levi form|80MTNXWW80]] of $\\partial\\Omega$ is positive semidefinite at every boundary point:\n$$\\sum_{j,k=1}^{n} \\frac{\\partial^2 \\rho}{\\partial z_j \\partial \\bar z_k}(p)\\, w_j \\bar w_k \\ge 0 \\quad \\text{for all } p \\in \\partial\\Omega \\text{ and all } w \\in H_p.$$\n\n## Remarks\n\nThe condition does not depend on the choice of $\\rho$, because another defining function multiplies the Levi form by a positive number.\nExample: every convex domain with $C^2$ boundary is Levi pseudoconvex.\nIndeed, for a convex domain the real Hessian of $\\rho$ at $p$ is positive semidefinite on the real tangent space of $\\partial\\Omega$ at $p$.\nFor $w \\in H_p$, both $w$ and $i w$ lie in this tangent space, and $L_\\rho(p; w)$ is one quarter of the sum of the real Hessian at $w$ and at $i w$.\nNon-example: the domain $\\set{z \\in \\C^2 : 1 \u003c \\abs{z_1}^2 + \\abs{z_2}^2 \u003c 4}$ fails at the inner sphere.\nThere $\\rho(z) = 1 - \\abs{z}^2$ is a defining function, and $L_\\rho(p; w) = -\\abs{w}^2 \u003c 0$ for $w \\in H_p$, $w \\ne 0$.\nWhen $n = 1$ the space $H_p$ is $\\set{0}$, so every planar domain with $C^2$ boundary is Levi pseudoconvex.\n",
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