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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 *strongly pseudoconvex* if the [[Levi form|80MTNXWW80]] of $\\partial\\Omega$ is positive definite at every boundary point:\n$$\\sum_{j,k=1}^{n} \\frac{\\partial^2 \\rho}{\\partial z_j \\partial \\bar z_k}(p)\\, w_j \\bar w_k \u003e 0 \\quad \\text{for all } p \\in \\partial\\Omega \\text{ and all } w \\in H_p \\setminus \\set{0}.$$\nA boundary point where this holds is a *strongly pseudoconvex point*.\n\n## Remarks\n\nA strongly pseudoconvex domain is [[Levi pseudoconvex|PETEQQCS0B]].\nMany authors also ask $\\Omega$ to be bounded.\nExample: the unit ball, with $\\rho(z) = \\abs{z}^2 - 1$, has Levi form $\\abs{w}^2$, which is positive for $w \\ne 0$.\nNon-example: the domain $\\set{z \\in \\C^2 : \\abs{z_1}^2 + \\abs{z_2}^4 \u003c 1}$ is Levi pseudoconvex but not strongly pseudoconvex.\nAt $p = (1, 0)$, with $\\rho(z) = \\abs{z_1}^2 + \\abs{z_2}^4 - 1$, the space $H_p$ is $\\set{w : w_1 = 0}$ and $L_\\rho(p; (0, w_2)) = 4 \\abs{p_2}^2 \\abs{w_2}^2 = 0$.\nA bounded domain $\\Omega$ with $C^2$ boundary is strongly pseudoconvex if and only if it has a defining function that is strictly plurisubharmonic near $\\partial\\Omega$.\nThis means that $L_\\rho(p; \\cdot)$ is positive definite on all of $\\C^n$, not only on $H_p$.\n",
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