[proofplan] We prove that the function $\delta_\Omega(z) := -\log d(z, \partial\Omega)$ is plurisubharmonic on a domain of holomorphy $\Omega$. The strategy uses the [Cartan--Thullen Theorem](/theorems/3385), which gives the distance characterisation $d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega)$ for all compact $K \subset \Omega$. We fix a complex line $\zeta \mapsto z_0 + \zeta w$ and verify the sub-mean-value inequality for $\psi(\zeta) := -\log d(z_0 + \zeta w, \partial\Omega)$ by applying the distance equality to the compact image of a closed disc, together with the maximum principle for holomorphic functions. [/proofplan] [step:Verify upper semicontinuity of $\delta_\Omega$] The boundary distance function $d(\cdot, \partial\Omega): \Omega \to (0, \infty)$ is continuous on $\Omega$ (in fact, $1$-Lipschitz with respect to the Euclidean metric, since $|d(z, \partial\Omega) - d(z', \partial\Omega)| \leq |z - z'|$). Since $d(\cdot, \partial\Omega) > 0$ on $\Omega$ and $\log$ is continuous on $(0, \infty)$, the function $\delta_\Omega = -\log d(\cdot, \partial\Omega)$ is continuous on $\Omega$. In particular, $\delta_\Omega$ is upper semicontinuous. [/step] [step:Fix a complex line and set up the sub-mean-value inequality using the distance characterisation] Fix $z_0 \in \Omega$ and $w \in \mathbb{C}^n$, and define \begin{align*} \psi: D &\to \mathbb{R} \\ \zeta &\mapsto -\log d(z_0 + \zeta w, \partial\Omega), \end{align*} where $D = \{\zeta \in \mathbb{C} : z_0 + \zeta w \in \Omega\}$ is open. We must show $\psi$ is subharmonic on $D$: for every $\zeta_0 \in D$ and $r > 0$ with $\overline{B}(\zeta_0, r) \subset D$, \begin{align*} \psi(\zeta_0) \leq \frac{1}{2\pi}\int_0^{2\pi} \psi(\zeta_0 + re^{i\theta})\,d\mathcal{L}^1(\theta). \end{align*} Define the compact set $K := \{z_0 + \zeta w : |\zeta - \zeta_0| \leq r\} \subset \Omega$ (compact because it is the continuous image of the closed disc $\overline{B}(\zeta_0, r)$, and contained in $\Omega$ because $\overline{B}(\zeta_0, r) \subset D$). Since $\Omega$ is a domain of holomorphy, the [Cartan--Thullen Theorem](/theorems/3385) implies holomorphic convexity, and the [Distance Characterisation of the Holomorphic Hull](/theorems/3402) gives \begin{align*} d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega). \end{align*} [guided] The sub-mean-value inequality for $\psi$ states that the value at the centre $\zeta_0$ is at most the average over the circle $|\zeta - \zeta_0| = r$. Since $\psi = -\log d(\cdot, \partial\Omega)$ and $-\log$ is decreasing, this is equivalent to showing that the boundary distance at the centre is at least as large as a suitable combination of boundary distances on the circle. The [distance characterisation of the holomorphic hull](/theorems/3402) is the key tool: it tells us that points in the hull $\hat{K}_\Omega$ are at least as far from $\partial\Omega$ as the closest point of $K$ to $\partial\Omega$. The compact set $K = \{z_0 + \zeta w : |\zeta - \zeta_0| \leq r\}$ is the image of the closed disc under the affine map $\zeta \mapsto z_0 + \zeta w$. This set is compact (continuous image of compact) and contained in $\Omega$ (since $\overline{B}(\zeta_0, r) \subset D$). The [Cartan--Thullen Theorem](/theorems/3385) guarantees that $\Omega$, being a domain of holomorphy, is holomorphically convex, and the [Distance Characterisation](/theorems/3402) then gives $d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega)$. [/guided] [/step] [step:Show that the centre point $z_0 + \zeta_0 w$ lies in the holomorphic hull $\hat{K}_\Omega$] We claim $z_0 + \zeta_0 w \in \hat{K}_\Omega$. By definition, $z_0 + \zeta_0 w \in \hat{K}_\Omega$ if and only if $|g(z_0 + \zeta_0 w)| \leq \sup_K |g|$ for every $g \in \mathcal{O}(\Omega)$. Fix any $g \in \mathcal{O}(\Omega)$. The function \begin{align*} \gamma: \overline{B}(\zeta_0, r) &\to \mathbb{C} \\ \zeta &\mapsto g(z_0 + \zeta w) \end{align*} is holomorphic on $B(\zeta_0, r)$ and continuous on $\overline{B}(\zeta_0, r)$ (since $g \in \mathcal{O}(\Omega)$ and the map $\zeta \mapsto z_0 + \zeta w$ is affine). By the [maximum modulus principle](/page/Maximum%20Modulus%20Principle) for holomorphic functions of one variable, $|\gamma|$ attains its maximum on the boundary: \begin{align*} |g(z_0 + \zeta_0 w)| = |\gamma(\zeta_0)| \leq \max_{|\zeta - \zeta_0| = r} |\gamma(\zeta)| = \max_{|\zeta - \zeta_0| = r} |g(z_0 + \zeta w)| \leq \sup_{z \in K} |g(z)|. \end{align*} Since this holds for every $g \in \mathcal{O}(\Omega)$, we conclude $z_0 + \zeta_0 w \in \hat{K}_\Omega$. [guided] This is the crucial link between the [maximum modulus principle](/theorems/491) and the holomorphic hull. The hull $\hat{K}_\Omega$ consists of all points $p \in \Omega$ satisfying $|g(p)| \leq \sup_K |g|$ for every $g \in \mathcal{O}(\Omega)$. To show that the centre point belongs to the hull, we need this inequality for every [holomorphic function](/page/Holomorphic%20Function) on $\Omega$. But restricting any such $g$ to the complex line gives a [holomorphic function](/page/Holomorphic%20Function) of one variable on the disc $B(\zeta_0, r)$, and the [maximum modulus principle](/page/Maximum%20Modulus%20Principle) in one variable immediately gives the bound. Note that the [maximum modulus principle](/theorems/491) is the reason the centre point lies in the hull of the disc's image. This is specific to holomorphic functions -- for arbitrary continuous functions, no such bound holds. [/guided] [/step] [step:Apply the distance characterisation to obtain the sub-maximum inequality for $\psi$] Since $z_0 + \zeta_0 w \in \hat{K}_\Omega$ (previous step) and $d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega)$ (distance characterisation): \begin{align*} d(z_0 + \zeta_0 w, \partial\Omega) \geq d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega) = \min_{|\zeta - \zeta_0| \leq r} d(z_0 + \zeta w, \partial\Omega). \end{align*} Since $d(\cdot, \partial\Omega)$ is continuous and $\overline{B}(\zeta_0, r)$ is compact, the minimum is attained. In particular, $d(z_0 + \zeta_0 w, \partial\Omega) \geq \min_{|\zeta - \zeta_0| = r} d(z_0 + \zeta w, \partial\Omega)$ (as the minimum over the full closed disc is at most the minimum over the boundary circle). Applying $-\log$ (which reverses the inequality since $-\log$ is decreasing): \begin{align*} \psi(\zeta_0) = -\log d(z_0 + \zeta_0 w, \partial\Omega) \leq -\log \min_{|\zeta - \zeta_0| = r} d(z_0 + \zeta w, \partial\Omega) = \max_{|\zeta - \zeta_0| = r} \psi(\zeta). \end{align*} This holds for every $\zeta_0 \in D$ and every $r > 0$ with $\overline{B}(\zeta_0, r) \subset D$. [/step] [step:Conclude subharmonicity from the sub-maximum property for continuous functions] The function $\psi$ is continuous on $D$ (established in the first step) and satisfies the sub-maximum property: $\psi(\zeta_0) \leq \max_{|\zeta - \zeta_0| = r} \psi(\zeta)$ for every closed disc $\overline{B}(\zeta_0, r) \subset D$ (established in the previous step). A continuous function $\psi: D \to \mathbb{R}$ on an [open set](/page/Open%20Set) $D \subset \mathbb{C}$ is subharmonic if and only if $\psi(\zeta_0) \leq \max_{|\zeta - \zeta_0| = r} \psi(\zeta)$ for every closed disc $\overline{B}(\zeta_0, r) \subset D$ (Ransford, *Potential Theory in the Complex Plane*, Theorem 2.3.1). Therefore $\psi$ is subharmonic on $D$, and in particular: \begin{align*} \psi(\zeta_0) \leq \frac{1}{2\pi}\int_0^{2\pi}\psi(\zeta_0 + re^{i\theta})\,d\mathcal{L}^1(\theta). \end{align*} Since $z_0 \in \Omega$ and $w \in \mathbb{C}^n$ were arbitrary, this shows that $\delta_\Omega = -\log d(\cdot, \partial\Omega)$ is plurisubharmonic on $\Omega$. [/step]