---
tag: Y71Z3MN0S3
kind: definition
name: Pseudoconvexity
msc: [32T05, 32U10]
defines: AHYZMSBBNH
uses: [FF2P7ER3QT, 0R6EWFKRCF, 02BH3MCBQB]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
    locator: Definition
---

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is *pseudoconvex* if there is a continuous [[plurisubharmonic function|FF2P7ER3QT]] $\varphi \colon D \to \R$ with the following property.
For every $c \in \R$, the set $\set{z \in D : \varphi(z) < c}$ has [[compact|0R6EWFKRCF]] closure contained in $D$.

A function $\varphi$ with this last property is called an exhaustion function of $D$.

## Remarks

Example: the unit ball $B = \set{z \in \C^n : \abs{z} < 1}$ with $\varphi(z) = -\log(1 - \abs{z}^2)$.
The 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.
The set $\set{\varphi < c}$ is the ball of radius $\sqrt{1 - e^{-c}}$ (empty for $c \le 0$), whose closure is compact and contained in $B$.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$.
Let $\varphi$ be any continuous plurisubharmonic function on $D$, and let $M$ be its maximum on the compact set $\set{\abs{z_1} \le 1/4, \ \abs{z_2} = 1/2} \subseteq D$.
For $0 < \abs{t} \le 1/4$, the disc $\zeta \mapsto (t, \zeta/2)$, $\abs{\zeta} \le 1$, lies in $D$, so by subharmonicity $\varphi(t, 0) \le M$.
The points $(t, 0)$ accumulate at the origin, which is not in $D$, so $\set{\varphi < M + 1}$ does not have compact closure in $D$, and no exhaustion of the required kind exists.

Equivalent 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.
If $D$ 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.